Lectures on Mean Curvature Flow (MAT 1063 HS) I. M. Sigal

Transcription

Lectures on Mean Curvature Flow (MAT 1063 HS)
I. M. Sigal
Dept of Mathematics, Univ of Toronto
November 23, 2014
Abstract
The mean curvature flow arises material science and condensed matter physics and has been recently
successfully applied to topological classification of surfaces and submanifolds. It is closely related to the
Ricci and inverse mean curvature flow.
The most interesting aspect of the mean curvature flow is formation of singularities, which is the
main theme of these lectures.
Background on geometry of surfaces and some technical statements are given in appendices.
These are rough, unedited notes, based on the course given a few years back and will changed and
further developed in the course of the present course. My thanks go to Greg Fournodavlos, Dan Ginsberg
and Wenbin Kong for the help with the material of the notes.
Contents
1 The mean curvature flow and its general properties
3
1.1
Mean curvature flow for level sets and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Special solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Different form of the mean curvature flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Mean curvature flow and the volume functional . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Finite time break-down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.6
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2 Self-similar surfaces and rescaled MCF
10
2.1
Solitons - Self-similar surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2
Rescaled MCF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3
Classification of mean-convex self-similar surfaces . . . . . . . . . . . . . . . . . . . . . . . . .
14
3 Global Existence vs. Singularity Formation
1
14
MCF Lectures, November 23, 2014
4 Normal hessians.
2
18
4.1
Spectra of hessians for spheres and cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.2
Connection of the spectrum and geometry of self-similar surfaces . . . . . . . . . . . . . . . .
22
4.3
Linear stability of self-similar surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.4
Explicit expression for the normal hessians . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.5
Hessians for spheres and cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.6
Minimal and self-similar submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.7
Translational solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5 Stability of elementary solutions I. Spherical collapse
28
5.1
Rescaled equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
5.2
The graph represetation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5.3
Reparametrization of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5.4
Lyapunov-Schmidt decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5.5
Linearized operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5.6
Lyapunov functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.7
Proof of Theorem 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
6 Stability of elementary solutions II. Neckpinching
36
6.1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
6.2
Rescaled surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6.3
Linearized map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6.4
Equations for the graph function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
6.5
Reparametrization of the solution space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
6.6
Lyapunov-Schmidt splitting (effective equations) . . . . . . . . . . . . . . . . . . . . . . . . .
42
6.7
General case. Transformed immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
6.8
Translational and rotational zero modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.9
Collapse center and axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.10 Equations for the graph function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
7 Volume preserving mean curvature flow
8 Volume preserving mean curvature flow for submanifolds:
48
Trapped surfaces
8.1
Variational formulation
3
52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
9 Ricci flow (to do)
61
10 Presentation topics
61
A Elements of Theory of Surfaces
62
A.1 Mean curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
A.2 Matrix representation of the first and second fundamental forms . . . . . . . . . . . . . . . .
64
A.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
A.4 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
A.5 Various differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
A.6 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
A.7 Riemann curvature tensor for hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
A.8 Mean curvature of normal graphs over the round sphere Sn and cylinder C n+1 . . . . . . . . .
71
B Maximum principles
B.1 Elliptic maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
B.2 Parabolic maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
C Elements of spectral theory
73
C.1 A characterization of the essential spectrum of a Schrödinger operator . . . . . . . . . . . . .
73
C.2 Perron-Frobenius Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
D Proof of the Feynmann-Kac Formula
1
73
77
The mean curvature flow and its general properties
The mean curvature flow, starting with a hypersurface S0 in Rn+1 , is the family of hypersurfaces S(t) given
by immersions ψ(·, t) which satisfy the evolution equation
∂ψ
= −H(ψ)ν(ψ)
∂t
(1.1)
ψ|t=0 = ψ0
where ψ0 is an immersion of S0 , H(x) and ν(x) are mean curvature and the outward unit normal vector
at x ∈ S(t), respectively. Recall that a map ψ : U → S, where U ⊂ Rn is open, is called the immersion,
4
iff dψ(u) is one-to-one (i.e. of the maximal rank) for every u ∈ U , and the mean curvature is defined as
H(x) = Tr dν(x), where ν : S → S n is the Gauss map given by ν : x → ν(x). (Wx := dν(x) : Tx S → Tν(x) S n
is the Weingarten map, or the shape operator. One can show that Wx : Tx S → Tx S, see Appendix A, where
the notions used above are defined and discussed.)
In this lecture we describe some general properties of the mean curvature flow, (1.1). We begin with
writing out (1.1) for various explicit representations for surfaces St .
1.1
Mean curvature flow for level sets and graphs
We rewrite out (1.1) for the level set and graph representation of S. Below, all differential operations, e.g.
∇, ∆, are defined in the corresponding Euclidian space (either Rn+1 or Rn ).
1) Level set representation S = {ϕ(x, t) = 0}. Then, by (A.2) of Appendix A, we have
∇ϕ
∇ϕ
,
H(x) = div
.
ν(x) =
|∇ϕ|
|∇ϕ|
∂ϕ
∂ϕ
∇ϕ
∇ϕ
∂x
We compute 0 = dϕ
dt = ∇x ϕ · ∂t + ∂t and therefore ∂t = ∇ϕ · |∇ϕ| div |∇ϕ| , which gives
∂ϕ
= |∇ϕ| div
∂t
∇ϕ
|∇ϕ|
(1.2)
.
(1.3)
2) Graph representation: S = graph of f over U ⊂ Rn . Here f : U → R. In this case S is the zero level
set of the function ϕ(x) = xn+1 − f (u), where u = (x1 , . . . , xn ) and x = (u, xn+1 ), and using (1.3) with
this function, we obtain
!
p
∂f
∇f
.
(1.4)
= |∇f |2 + 1 div p
∂t
|∇f |2 + 1
Denote by Hess f the standard euclidean hessian, Hess f :=
∂2f
∂ui ∂uj
. Then we can rewrite (1.4) as
∂f
∇f Hess f ∇f
= ∆f −
.
∂t
|∇f |2 + 1
(1.5)
We can also consider graphs over another surface, say, the sphere, S n , i.e. ψ : S n → Rn+1 , with ψ given
by
ψ(ω, t) = f (ω, t)ω, where f : S n → R,
or the cylinder, C n = R × S n , i.e. ψ : C n → Rn+1 , with ψ given by
ψ(x, ω, t) = (x, f (x, ω, t)ω), where f : C n → R
1.2
5
Special solutions
Static solutions: Static solutions satisfy the equation H(x) = 0 on S, which, by Proposition 2 below, is the
equation for critical points of the volume functional V (ψ) (the Euler - Lagrange equation for V (ψ)). Thus,
static solutions of the MCF (1.1) are minimal surfaces.
There is a considerable literature on minimal surfaces (see e.g. [59, 18, 19]). Much of it related to the
Bernstein conjecture:
The only entire minimal graphs are linear functions.
It was shown it is true for n ≤ 7:
(a) Bernstein for n = 2;
(b) De Georgi for n = 3;
(c) Almgren for n = 4; .
(d) Simons for n = 5, 6, 7.
(b) and (d) used in part results of Fleming on minimal cones. These results were extended by Schoen, Simon
and Yau.
Bombieri, De Georgi and Giusti constructed a contra example for n > 7.
Explicit solutions:
a) Sphere x(t) = R(t)ˆ
ˆ = x/kxk, or {ϕ(x, t) = 0}, where ϕ(x, t) := |x|2 − R(t)2 , or graph f ,
p x, where x
where f (u, t) = R(t)2 − |u|2 . (SR = RS n , where S n is the unit n-sphere.) Then we have
n
∇ϕ
= div(ˆ
x) =
H(x) = div
|∇ϕ|
R
p
n
and therefore we get R˙ = − R
which implies R = R02 − 2nt. So this solution shrinks to a point.
n−1
˙
b) Cylinder x(t) = (R(t)xˆ0 , x00 ), where x = (x0 , x00 ) ∈ Rk+1 × Rm . Then H(x) = n−1
R and R = − R
p
which implies R = R02 − 2(n − 1)t. (In the implicit function representation the cyliner is given by
P
1
n−1 2 2
.)
{ϕ = 0}, where ϕ := r − R, with r =
x
i=1 i
Not explicit example: Motion of torus (H. M. Soner and P. E. Souganidis).
1.3
Different form of the mean curvature flow
Multiplying the equation (1.1) in Rn+1 by ν(ψ), we obtain the equation
hν(ψ),
∂ψ
i = −H(ψ).
∂t
(1.6)
6
In opposite direction we have
Proposition 1. (i) The equations (1.1) and (1.6) are equivalent in the sense that if ψ satisfies (1.6), then
it satisfies (1.1) and, if ψ satisfies (1.6), then there is a (time-dependent) reparametrization ϕ of S, s.t.
ψ ◦ ϕ satisfies (1.1). (ii) The equation (1.6) is invariant under reparametrization. (One can say that (1.1)
involves fixing a specific parametrization (fixing the gauge).)
∂ψ
∂x
T
Proof. Denote ( ∂ψ
:= ∂ψ
˙ =
∂t )
∂t − hν, ∂t iν (the projection of ∂t onto Tψ S) and let ϕ satisfy the ODE ϕ
∂
∂
T
◦
ϕ)
(parametrized
by
u
∈
U
).
Then
(ψ
◦
ϕ)
=
(
ψ)
◦
ϕ
+
dψ
ϕ.
˙
Substituting
ϕ
˙
=
−(dψ)−1 ( ∂ψ
∂t
∂t
∂t
∂ψ
∂ψ
∂ψ N
∂ψ T
∂ψ N
T
T
−(dψ)−1 ( ∂ψ
◦
ϕ)
into
this
and
using
that
hν,
(
◦
ϕ)
i
=
0
and
=
(
)
+
(
)
,
where
(
)
=
∂t
∂t
∂t
∂t
∂t
∂t
∂ψ
∂
∂
hν, ∂t iν, we obtain ∂t (ψ ◦ ϕ) = hν, ( ∂t ψ) ◦ ϕiν ◦ ϕ = −(Hν) ◦ ϕ.
To show that the equation (1.6) is invariant under reparametrization, we observe that, by the above,
∂
∂
hν ◦ ϕ, ( ∂t
ψ) ◦ ϕ + dψ ◦ ϕ ϕi
˙ = hν, ( ∂t
ψ)i ◦ ϕ.
1.4
Mean curvature flow and the volume functional
We show that the mean curvature arises from the first variation of the surface area functional. We do this
for surfaces which are graphs of some function, which is locally the case for any surface.
Recall that locally the surface volume functional can be written as
Z
√ n
V (S ∩ U ) =
gd u,
U
or V (ψ) =
functional.
R √
ψ
gdn u, where g := det(gij ). We show that the MCF is a gradient flow of the surface area
Recall that, if E : M → R is a functional on an infinite dimensional manifold M , then the Gâteaux
derivative dE(u), u ∈ M , of E(u), is a linear functional on X defined as
dE(u)ξ =
∂
E(us )|s=0 ,
∂λ
(1.7)
where us is a path on M satisfying us |s=0 = u and ∂s us |s=0 = ξ, with ξ ∈ Tu M , if the latter derivative
exists.
If M is a Riemannian manifold, with an inner product (Riemann metric) h(ξ, η), then there exists a
vector field, gradg E(u), such that
h(gradh E(u), ξ) = dE(u)ξ.
(1.8)
This vector is called the gradient of E(u) at u w.r.t. the metric h.
We want to compute the Gˆ
ateaux derivative of the area functional V . Any infinitesimal variation
(vector field on the space of surfaces), η, can be decomposed into the normal and tangential components,
η = η N + η T , where η N is the projection onto the normal vector-filed ν, η N = hν, ηiν. As we have shown,
the tangential component, η T , generates a reparametrization of the surface. Since V is independent of
reparametrization, it suffices to look only at normal variations, ψs , of the immersion ψ, i.e. generated by
vector fields η, directed along the normal ν: η = f ν. We begin with
7
Proposition 2. For a surface S given locally by an immersion ψ and normal variations, η = f ν, we have
Z
√
dV (ψ)η =
Hν · η gdn u.
(1.9)
U
Proof. We want to show for g ≡ g(ψ) := det(gij ) that
√
√
d gη = H gν · η.
(1.10)
To this end we denote gˆ := (gij ) and use the representation det gˆ = eTr ln gˆ , which gives dgη = g Tr[ˆ
g −1 dˆ
g η]
and therefore
dgη = gg ji dgij η.
D
E
D
E
D
∂ψ ∂η
∂ψ
∂
∂
Next, we compute dgij η = 2 ∂u
= 2 ∂u
i , ∂uj
j
∂ui , η − 2 ∂uj
(1.11)
E
∂ψ
∂ui , η . For normal variations, η = f ν, the
2
ψ
last relation and the relation bij = −h ∂u∂i ∂u
, νi (see Lemma 57 of Appendix A) give
j
dgij η = 2bij (ν · η),
(1.12)
where bij are the matrix elements of the second fundamental form. This relation, together with the relation
dg(ψ)η = g(ψ)g ji dgij η, proven above, and, implies dgη = 2(ν · η)gg ji bij , which, due to the relation bij =
2
ψ
, νi, gives
−h ∂u∂i ∂u
j
dgη = 2gH(ν · η).
√
1 −1/2
This, together with d gη = 2 g
dgη, gives in turn (1.10).
Corollary 3. Critical points of V (S), which are called the minimal surfaces satisfy the equation (the EulerLagrange equation for V (S)) H(x) = 0.
We define the normal gradient, gradN V (ψ), as grad V (ψ), restricted to the normal vector fields, ξ =
f ν. Then the equation (1.9) and the definition (1.8) imply that the (normal) gradient of V (ψ) at ψ is
gradN V (ψ) = H(ψ)ν(ψ), which implies the following
R
Theorem 4. The mean curvature flow is a gradient flow with the functional V and the metric S ξ · η:
∂t ψ = − gradN V (ψ).
Corollary 5. If S evolves according to (1.1), then V (S) decreases. In fact, ∂t V (S) = −
R
S
H 2 (x) < 0.
Proof. The assertion on ψ(U ) follows from (1.9) as
Z
Z
∂t V (ψ) =
Hν · ∂t ψ = − H 2 (x),
U
ψ
which after using a partition of unity proves the statement.
As a result, the area of a closed surface shrink under the mean curvature flow. Note also that the
equation (1.12) implies ∂t gij = 2bij (ν · ∂t ψ), which, together with (1.6), gives the following equation for the
evolution of the metric
∂t gij = −2bij H.
(1.13)
8
Graph representation. Note that, if S = graph f, f : U → R, then we can take ψ(u) = (u, f (u)) and
the proof of (1.9) simplifies. In this case we can take ψ(u) = (u, f (u)) and the surface area functional is
given by
Z p
V (f ) =
1 + |∇f |2 dn u.
U
we compute, using that ξ|∂U = 0,
∇f
Z
· ∇ξdn u,
1 + |∇f |2
which, after integrating by parts and using the equation H(x) = − div √ ∇f 2 (see (A.3) of Appendix
dV (f )ξ = ∂s |s=0 V (f + sξ) =
U
p
1+|∇f |
A), gives
Z
dV (f )ξ =
Hξdn u.
(1.14)
U
Using (1.14) we find
Z
dV (f )ξ = −
Hξd u = −
U
So we have grad V (f ) = −
p
n
Z p
U
1 + |∇f |2 Hξ p
1 + |∇f |2 H(x) in the metric hf (ξ, η) :=
dn u
1 + |∇f |2
R
U
ξη √
.
dn u
,
1+|∇f |2
which together with
(1.4) gives ∂t f = − grad V (f ).
1.5
Finite time break-down
The following general result shows, using the maximum principle, that, under the MCF, the diameter of the
closed surface shrinks to 0 in finite time:
Theorem 6. If S0 ⊂ Bρ0 (compact and without boundary?) for some ρ0 > 0, where Bρ is the ball with
ρ2
radius ρ, center at 0, then St ⊂ B√ 2
. In particular, (MCF) exists for time t∗ < 0 .
2n
ρ0 −2nt
∂f
Proof of Theorem 6. Let f (x, t) = |x|2 + 2nt. Compute df
dt = ∂t + grad f ·
→
−
H = −Hν. We also have
k
√
∆S |x|2 = √1g ∂ui ( gg ij 2xk ∂x
∂uj )
k
√
∂xk
= 2xk ∆S xk + √2g gg ij ∂x
∂ui ∂uj
→
−
= 2x · H + 2g ij gij
→
−
= 2x · H + 2n.
So we obtain
df
dt
∂x
∂t
→
−
= 2n + 2x · H , where
= ∆S f .
d
Let f˜ = f − t for > 0, then ( dt
− ∆S )f˜ = − < 0. We claim that maxSt f˜ ≤ maxS0 f . Otherwise let t∗
˜
be the first time when maxSt f reaches a value larger than maxS0 f and f˜(x∗ ) = maxSt f˜. Then at (x∗ , t∗ ),
9
2
∂ f˜
df
jk ∂ f˜
dt f˜ ≤ 0, ∂ui f˜ = 0 and (∂ui ∂uj f˜) ≤ 0. Hence ∆S f˜ = (∂uj + Γiij )g jk ∂u
k = g
∂uj ∂uk ≤ 0. So dt − ∆S f ≥ 0,
contradiction. This proves our claim maxSt f˜ ≤ maxS0 f ∀ > 0. Let → 0, we have maxSt f ≤ maxS0 f ,
i.e. |x|2 ≤ ρ20 − 2nt at time t.
How this collapse take place? Some indications come from the the results of Section 3 below.
1.6
Symmetries
Proposition 7. Mean curvature is invariant under translations (S → S + h ∀h ∈ Rn+1 ), rotations (S →
g −1 S ∀g ∈ O(n + 1)) and scaling transformations,
S(t) → λS(λ−2 t)
⇔
x→
λx,
t → λ−2 t.
The invariance under translations and rotations is obvious. To prove the invariance under scaling, we
first show that
H(λx) = λ−1 H(x).
Indeed, looking at the definitions bij =
∂2ψ
∂ui ∂uj
bij → λbij
· ν and gij :=
D
and gij → λ2 gij
(1.15)
∂ψ ∂ψ
∂ui , ∂uj
if
E
, we see that
ψ → λψ.
Hence g ij bji → λ−1 g ij Bji , which due to the equation H = g ij bji (see Lemma 57 of Appendix A), implies
(1.15).
Now, let τ := λ−2 t, and ψ λ (t) := λψ(τ ). Then
∂t ψ λ = λλ−2 (∂τ ψ)(τ ) = −λ−1 H(x(τ ))ν(ψ(τ )).
On the other hand, ν λ ≡ ν(ψ λ (τ )) = ν ≡ ν(ψ(τ )) and H λ ≡ H(ψ λ (τ )) = λ−1 H(ψ(τ )) ≡ λ−1 H (by (1.15)),
which gives
∂t ψ λ = −H λ ν λ .
(1.16)
We can also use the scaling ψ(u, t) → λψ(λ−1 u, λ−2 t), together with the fact that H → H, under
u → λ−1 u. Indeed, bij → λ−2 bij and gij → λ−2 gij if ψ(u) → ψ(λ−1 u), and therefore g ij bji → g ij bji .
For S = graph f , f (u) → λf (λ−1 u) ⇔ ψ(u) → λψ λ−1 u , where ψ(u) := (u, f (u)). Indeed,
ψ(u) := (u, f (u)) → u, λf (λ−1 u) = λ λ−1 u, f (λ−1 u) =: λψ λ−1 u .
Spherically symmetric and axi-symmetric (equivariant) solutions:
Convexity
To-do
2
10
Self-similar surfaces and rescaled MCF
2.1
Solitons - Self-similar surfaces
Consider solutions of the MCF of the form S(t) ≡ S λ(t) := λ(t)S (standing waves), or ψ(u, t) = λ(t)ϕ(u),
˙ = −λ−1 H(ϕ)ν(ϕ). Multiwhere λ(t) > 0. Plugging this into (1.1) and using H(λy) = λ−1 H(y), gives λϕ
plying this by λ and ν(ϕ), we obtain
H(ϕ) = ahν(ϕ), ϕi,
(2.1)
and λλ˙ =p−a. Since H(ϕ) is independent of t, then so should be λλ˙ = −a. Solving the last equation, we
find λ = λ20 − 2at.
i) a > 0 ⇒ λ → 0 as t → T :=
λ20
2a
⇒ S λ is a shrinker.
ii) a < 0 ⇒ λ → ∞ as t → ∞ ⇒ S λ is an expander.
iii) a = 0 ⇒ λ =const and S λ is a minimal surface.
The equation (2.1) has the solutions: a is time-independent and x is one of the following
a) Sphere x = Rˆ
x, where R =
pn
a.
b) Cylinder x = (Rxˆ0 , x00 ), where x = (x0 , x00 ) ∈ Rk+1 × Rm , where R =
q
k
a.
As stated in the following theorem, these solutions are robust.
Theorem 8 (Huisken). Let S satisfy H = ax · ν, a > 0, and H ≥ 0. We have
p
(i) If n ≥ 2, and S is compact, then S is a sphere of radius na .
(ii) If n = 2 and S is a surface of revolution, then S is the cylinder of radius
q
n−1
a .
A surface S given by an immersion ϕ will be called the self-similar surface.
Now we consider self-similar surfaces in the graph representation. Let
f (u, t) = λχ(λ−1 u),
λ depends on
˙ we find
Substituting this into (1.4) and setting y = λ−1 u and a = −λλ,
q
1 + |∇y χ|2 H(χ) = a(y∂y − 1)χ.
t.
(2.2)
(2.3)
We see that the minimal surface equation H(ϕ) = 0 is a special case of (2.1) and minimal surfaces form
a special class of self-similar surfaces.
11
Translation solitons. These are solutions of the MCF of the form S(t) ≡ S + h(t) (traveling waves),
or ψ(u, t) = ϕ(u) + z(t), where h(t) ∈ Rn+1 . Plugging this into (1.1) and using H(y + z) = H(y), gives
h˙ = −H(ϕ)ν(ϕ), or
H(ϕ) = −hν(ϕ), vi,
h˙ = v.
and
(2.4)
Since H(ϕ) is independent of t, then so should be z˙ = v. The translation solitons are static solutions of the
equation
(∂t ϕ)N = −H(ϕ) − v N ,
(2.5)
where v N := hv, ν(ϕ)i, for ϕ and z. (more to come)
Breathers.
2.2
MCF periodic in t.
Rescaled MCF.
First we note that the self-similar surfaces are static solutions of the rescaled MCF,
(∂τ ϕ)N = −H(ϕ) + aϕN ,
(2.6)
˙ which is obtained by rescaling the surface ψ
where (∂τ ϕ)N := h∂τ ϕ, ν(ϕ)i, ϕN := hϕ, ν(ϕ)i and a = −λλ,
and time t as
Z t
ϕ(u, τ ) := λ−1 (t)ψ(u, t), τ =
λ−2 (s)ds.
(2.7)
0
˙ −2 ψ + λ−1 ∂t ψ), (2.7), (1.1), λH(ϕ) = H(λ−1 ϕ) and ν(ψ) = ν(ϕ), we
Indeed, using ∂τ ϕ = λ2 ∂t ϕ = λ2 (−λλ
find
˙
∂τ ϕ = −λλϕ
− λH(ψ)ν(ψ) = aϕ − H(ϕ)ν(ϕ).
Multiplying this by ν(ϕ) gives (2.6).
˙ = −a, we obtain the parabolic scaling:
For static solutions, a =const. Then solving the equation λλ
λ=
p
2a(T − t)
and τ (t) = −
1
ln(T − t),
2a
(2.8)
where T := λ20 /2T , which was already discussed in connection with the scaling solitons.
Note that the equation (2.1) for self-similar surfaces is the equation for static solutions to (2.6). (This
is another analogy with minimal surfaces.)
Now, we know that minimal surfaces are critical points of the volume functional V (ψ) (by Proposition
2, the equation H(x) = 0 on S is the Euler - Lagrange equation for V (ψ)). Are self-similar surfaces
critical points of some modification of the volume functional? (Recall that, because of reparametrization
(see Proposition 1), it suffices to look only at normal variations, ψs , of the immersion ψ, i.e. generated by
vector fields η, directed along the normal ν: η = f ν.) The answer to this question is yes and is given in the
following
12
R
2
a
Proposition 9. Let ρ(x) = e− 2 |x| and Va (ϕ) := S λ ρ (conformal area of S λ ). For a surface S given locally
by an immersion ψ and normal variations, η = f ν, we have
Z
dVa (ϕ)η =
(H − aϕN )hν, ηi ρ.
(2.9)
Sλ
p
R
2
a
Proof. The definition of Va (ϕ) gives Va (ϕ) = U ρ(ϕ) g(ϕ)dn u, where ρ(ϕ) = e− 2 |ϕ| . We have dρ(ϕ)η =
√
√
−aρ(ϕ)hν, ηi. Using this formula and the equation d gη = H ghν, ηi proven above (see (1.10)) and the
fact that we are dealing with normal variations, η = f ν, we obtain (2.9).
By the definition (1.8), the definition of the normal gradient and the formula (2.9), we have
N
gradN
h Va (ϕ) = (H − aϕ )ν
R
in the Riemann metric h(ξ, η) := S λ ξ · ηρ, where ξ · η = hξ, ηi is the Euclidean inner product in Rn+1 . This
and the equation (2.6) and Proposition 9 imply
Corollary 10. Assume a in (2.6) is constant (i.e. the rescaling (2.7) is parabolic, (2.8)). Then
a) Assume a is constant. Then the modified area functional Va (ϕ) is monotonically decaying under the
rescaled flow (2.6), more precisely,
Z
∂τ Va (ϕ) = −
ρ|H − aϕN |2 .
(2.10)
Sλ
b) The renormalizedRflow (2.6) is a gradient flow for the modified area functional Va (ϕ) and the Riemann
metric h(ξ, η) := S λ ξ · ηρ:
ϕ˙ = − gradh Va (ϕ).
The relation (2.10) is the Huisken monotonicity formula (earlier results of this type were obtained by
Giga and Kohn and by Struwe).
Most interesting minimal surfaces are not just critical points of the volume functional V (ψ) but are
minimizers for it. What about self-similar surfaces? We see that
(i) inf Va (ϕ) = 0 and Va (ϕ) is minimized by any sequence shrinking to a point.
(ii) Va (ϕ) is unbounded from above. This is clear for a < 0. To see this for a > 0, we construct a sequence
of surfaces lying inside a fixed ball in Rn+1 and folding tighter and tighter.
Thus self-similar surfaces are neither minimizers nor maximizers of Va (ϕ). We conjecture that they are
saddle points satisfying min-max principle: supV inf ϕ:V (ϕ)=V Va (ϕ). One can try use this principle (say in
the form of the mountain pass lemma) to find solutions of (2.1).
13
Appendix: Graph representation. As an exercise, we prove (2.9) for a local patch, S λ ∩ W given by a
graph of a function f λ : U λ → R, S λ ∩ W = graph f λ which is the rescaling
Z t
λ−2 (s)ds,
(2.11)
f λ (v, τ ) = λ−1 f (u, t), v = λ−1 u, τ =
0
p
p
˙ λ − λy
˙ · ∇v f λ f λ +
of S =graph f . In this case, using that 1 + |∇f λ |2 λ−1 H λ = 1 + |∇f |2 H and ∂t f = λf
−2
λ
λ
λλ ∂τ f , we find that f (v, τ ) satisfies the equation
!
q
λ
∇f
λ
− a(v∂v − 1)f λ .
(2.12)
∂τ f = 1 + |∇f λ |2 div p
1 + |∇f λ |2
In the rest of the derivation we omit the superindex λ. Let x = (v, f (v)). Since |x|2 = |v|2 + |f (v)|2 , we can
write in local coordinates
Z
Z
2
2 2p
a
Va (f ) :=
ρ=
e− 2 (v +f ) 1 + |∇f |2 dn v.
ψ
We compute dVa (f )ξ =
R
U
ρ −af ξ +
U
√∇f ·∇ξ 2
1+|∇f |
. Integration by parts of the second term gives
"
p
∇f
dVa (f )ξ =
ρ −af 1 + |∇f |2 + a(v + f ∇f ) p
− div
λ
1
+
|∇f |2
U
Z
!#
∇f
ξdn v.
p
1 + |∇f |2
The first two terms on the
from differentiating ρ. Reducing them to the common denominator
r.h.s. come
∇f
, the mean curvature, we find
and using that H = div √
2
1+|∇f |
(v · ∇ − 1)f
dVa (f )ξ = −
ρ H − ap
1 + |∇f |2
U
Z
!
ξdn v.
(2.13)
−v·∇f +f
Now ν = √(−∇f,1) 2 and x = (v, f (v)). Hence we have ν · x = √
, which gives (2.9).
2
1+|∇f |
1+|∇f |
As before, (2.13) implies
R that (2.12) is the gradient flow for the functional Va :=
and the metric h(ξ, η) := U λ ξη √ ρ λ 2 :
1+|∇f |
f˙λ = − gradh Ja (f λ ).
Indeed, he equation (2.13) implies
(y · ∇ − 1)f λ
gradh Va (f ) = − H − p
1 + |∇f λ |2
λ
which gives the result.
!q
1 + |∇f λ |2 .
a
R
Uλ
e− 2 (v
2
+f 2 )2
p
1 + |∇f |2 dn v
2.3
14
Classification of mean-convex self-similar surfaces
Theorem 11 (Huisken ([43, 44]), Colding - Minicozzi (removed Huisken’s restriction on the boundedness
of |A| in the non-compact case)). The only smooth, complete, embedded self-similar surfaces in Rn+1
q, with
a > 0, polynomial growth and H(ϕ) > 0, are S k × Rn−k , where S k is the round k−sphere of radius
k
a.
Some steps of the proof in the compact case following [43]. Let {ej } be an orthonormal frame on S and ∇j
denote the covariant derivatives w.r.to {ej }. As usual W denotes the Weingarten map and |W | its HilbertSchmidt norm, |W |2 := Tr(W 2 ). Differentiating the self-similarity equation H(ϕ) = hν(ϕ), ϕi, where we set
a = 1, one derives the following equations
(a) ∆H = H − H|W |2 + hϕ, ej i∇j H.
(b) ∆f = 2H −4 |T |2 − vi ∇i f , where f := |W |2 /H 2 , Tijk := bij ∇k W − ∇k bij W and vi := 2H −1 ∇i H −
hϕ, ej i.
By the second equation, since S is compact, we see by the maximum principle (see Appendix B.1) that f
is a constant. Hence there is a number α s.t. |W |2 = α/H 2 . Moreover, T = 0. Decomposing T into the
symmetric and antisymmetric parts, T sym and T anti−sym , and using that |T |2 = |T sym |2 + |T anti−sym |2 , we
anti−sym
conclude that T anti−sym = 0. Now writing out Tijk
= 12 (bij ∇k W − bik ∇j W − (∇k bij W − ∇j bik W ))
and using Codazzi’s equality ∇k bij ∇k = ∇j bik , we find bij ∇k W − bik ∇j W = 0. (to continue)
Compare with the classification of minimal surfaces, specifically the Bernstein conjecture, mentioned
above.
3
Global Existence vs. Singularity Formation
LWP and the maximal interval of existence; singularities of type I, Huisken, [40]
The next two results indicate some possible scenarios for the long-time behavior of surface under mean
curvature flow,
∂ψ
= −H(ψ)ν(ψ)
∂t
(3.1)
ψ|t=0 = ψ0
Definition: S is uniformly convex if all curvatures of S ≥ δ for some δ > 0, i.e. the Weingarten map, W , is
strictly positive-definite.
Theorem 12. (Ecker-Huisken) If S0 is an entire graph over a hyperplane in Rn+1 , satisfying certain growth
conditions, then the MCF St , starting at S0 , exists ∀t and converges to a a hyperplane.
Theorem 13. ( Huisken [40]) Let S0 be a compact p
uniformly convex hypersurface. Then there is T ∈ (0, ∞)
s.t. (for a sequence tn → T ) λ−1 S → S∞ , for λ = 2n(T − t), where S∞ is a sphere.
15
In the both cases, the rescaled flow converges to a surface S∞ , which satisfies H = −ν · x in the first
case and H = ν · x in the second.
The next result establishes a general relation between the bound on the singularity, supS |W |2 ≤ T c−t
(Type I collapse), and the behaviour near collapse time. Let W be the Weingarten map of the surface and
let |W |2 := T r(W 2 ) = g ij g kl bik bjl , where recall bik are the matrix elements of W in the natural basis, i.e.
(bik ) is the second fundamental form. (Thus |W |2 is the square root of the sum of squares of principal
curvatures.)
Theorem 14. (Huisken [43]) Let n ≥ 2, S0 be a compact and T > 0 be the maximal existence time for the
MCF stsrting with S0 . Assume the solution to the MCF satisfies
c
.
(3.2)
sup |W |2 ≤
T
−t
S
p
Then there is T ∈ (0, ∞) s.t. (for a sequence tn → T ) λ−1 S → S∞ , for λ = 2n(T − t), where S∞ satisfies
H = ν · x.
Now, a general result given in Theorem 6 above shows, using the maximum principle, that, under the
MCF, the diameter of the closed surface shrinks to 0. The last two theorems and their proofs indicate how
this collapse take place.
Main steps of proof of Theorem 14. We conduct the proof in five steps.
Step #1: Rescaling. To understand the asymptotic behavior of the surface, we rescale it so that the rescaled
surface converges to a limit (similarly as in traveling wave and blowup problems). Consider the new surface
S λ given by (blowup variables) (2.7),
Z t
ϕ(u, τ ) := λ−1 (t)ψ(u, t), τ =
λ−2 (s)ds.
(3.3)
0
Its immersion ϕ satisfies the equation
∂τ ϕ = −H(ϕ)ν(ϕ) + aϕ,
˙ We take λ =
where a = −λλ.
(3.4)
p
2n(T − t) (so that a = n.). We prove some a priori estimates on the flow.
p
Step #2: |W |-bound. By the condition supS |W |2 ≤ T c−t , the rescaling λ−1 S → S∞ , with λ = 2n(T − t),
and the scaling relation W (λ−1 ψ) = λW (ψ) we have that the Weingarten map W λ of the rescaled surface
S λ satisfies
sup |W λ | ≤ c0 .
(3.5)
S
Step #3: Lyapunov functional. On this step one uses Huisken monotonicity formula (2.10), which we recall
here
Z
Z
∂τ
ρ=−
ρ|H − aν · ϕ|2 ,
(3.6)
Sλ
2
−a
2 |x|
Sλ
2
−a
2 |x−x0 |
. Let ρx0 (x) = e
. This relation, together with the inequality V (S λ ∩ BR (x0 )) ≤
with ρ(x) = e
R
2
eaR /2 S λ ρx0 , implies the following estimate: for any x0 ,
Z
2
V (S λ ∩ BR (x0 )) ≤ eaR /2
ρx0 |t=0 .
(3.7)
Sλ
16
Step #4: Compactness. Let x∗ ∈ Rn+1 be the collapse point of the MCF ψ, i.e. the point such that
ψ(u, t) → x∗ , as t → T , for some u ∈ U . Then, we have by (3.1) and (3.2) that
Z
T
|ψ(u, t)| ≤
|H(u, t)|dτ ≤ C(T − t)1/2 .
t
Then, by (3.3), with λ =
p
2n(T − t), we have
|ϕ(u, τ )| ≤ C.
(3.8)
Step #5: Compactness. By the local existence theorem there T > 0, s.t. the MCF exists on the interval
[0, T ). Assume that T is the maximal
p existence time. Let T < ∞. We want to show that there is a sequence
tn → T s.t. λ−1 S → S∞ , for λ = 2n(T − t). To this end, we use the following key result:
Theorem 15 (Compactness theorem: Langer, n = 3). Given constants A > 0, E > 0, p > n, the set Ω of
immersed surfaces ψ : S → Rn satisfying
• V (ψ) =
R
S
dvol < A;
• Ep (ψ) := S Tr |W |p dA < E;
R
• S ψdvol = 0 (center of gravity at 0);
R
is compact in the sense that ∀ {ψ n } ⊂ Ω, ∃ surface diffeomorphisms αn such that a subsequence of {ψ n ◦ αn }
converges in the C 1 topology to an immersion ψ in Ω. Here S is compact and without boundary.
The exponential p is determined by the Sobolev embedding W 2,p (Rn ) ⊂ C 1 (Rn ), where W s,p (Rn ) is the
Sobolev space of functions integrable in p power together with their derivatives up to the order s.
By Langer’s theorem, the estimates (3.5) and (3.7) imply that the family S λ (τ ) ∩ BR (x0 ), 0 ≤ τ < T,
is compact (up to the closure) for each R > 0 in the C 1 topology. Hence S λ (τ 0 ) ∩ BR (x0 ) converges, for a
subsequence {τ 0 }, τ 0 → T. By (3.8), for any R sufficiently large, the limit is non-trivial. Hence, by picking a
diagonal sequence, we find that S λ (τ 00 ) converges, as τ 00 → T. Denote the limit by S λ (T ). We can take it for
the new initial condition and continue the flow beyond T . This leads to the contradiction. Hence T = ∞.
Step #5: Identification of S∞ . To show that if S λ (τ ) → S∞ , then S∞ satisfies
H(y) = aν · y,
(3.9)
one uses the monotonicity formula. Indeed, integrating (3.6), we obtain
Z ∞ Z
Z
Z
Z
dτ
ρ|H − aν · x|2 =
ρ|τ =0 −
ρ|τ =∞ ≤
0
This gives
R
Sλ
Sλ
Sλ
Sλ
ρ|τ =0 .
Sλ
ρ|H λ − aν λ · xλ |2 → 0 as τ → ∞. So we get (3.9).
If n ≥ 2, H ≥ 0,p
and if S compact and satisfies (3.9), then, by the Huisken classification theorem, S is
the sphere of radius na , we conclude that S is a sphere in the second case.
17
Remark.PAnother way to show that it converges to the round sphere, one estimates the difference |W |2 −
1
1
2
2
2
2
i<j (κi − κj ) , where κj are the principal curvatures. (Note that H ≤ n|W | .) One shows that
nH = n
there is δ > 0 s.t. |W |2 − n1 H 2 ≤ Cδ H 2−δ . After rescaling we obtain
|W λ |2 −
1 λ 2
(H ) ≤ Cδ (H λ )2−δ λδ .
n
Hence, if |H λ | is uniformly bounded, then |W λ |2 − n1 (H λ )2 → 0, as t → T , and therefore the rescaled surface
converges to a sphere.
Now, by a local existence result we know that St exists on some time interval. Let [0, T ) be the maximal
time interval for the existence of the MCF. Proceeding as in the proof of Theorem 14, one shows the following
important result, which shows that limt→T supS |W | is the quantity that controls the global existence or blow
up.
Theorem 16. (Huisken [40]) T < ∞ ⇐⇒ limt→T supS |W | = ∞.
We estimate the maximal existence (collapse) time, T . Specifically, we want to prove the following
2
)−1 < ∞.
Theorem 17. Let n ≥ 2, S0 is a closed surface and min H > 0 on S0 . Then T ≤ ( n2 Hmin
Recall that bij denote the matrix elements of the second fundamental form. We begin with
Theorem 18 (Evolution of geometric quantities). We have
• ∂t g ij = 2Hbij and
• ∂t bij = ∇i ∇j H − Hbik bkj .
In (1.13) we have shown that ∂t gij = −2bij H (which follows from (1.12) (dgij η = 2bij (ν · η))). The first
relation of the theorem can be shown similarly or using this relation. For the second relation, see [40] or [22],
App B. (One can also derive it from statement (b) of Lemma 31 below, i.e. dbij (ϕ)η = −hν, (∂i ∂j + Γkij ∂k )ηi
and the properties of the connections given in Appendix A.4.) Theorem 18 implies the following differential
inequality for the mean curvature
Corollary 19. ∂t H ≥ ∆H + n1 H 3 .
Indeed, the definition H = g ij bij gives ∂t H = ∂t g ij bij + g ij ∂t bij . Applying the relations of Theorem 18
gives ∂t H = 2Hbij bij + g ij (∇i ∇j H − Hbik bkj ). Now, using the relations ∆bij = ∇i ∇j H + Hbik bkj − |W |2 bij
(see [40]) and |W |2 = bij bij and the definition H = g ij bij , we find
∂t H = ∆H + H|W |2 .
Finally, the inequality |W |2 ≥
1
2
nH
completes the proof. The following result implies Theorem 17.
Theorem 20. Let n ≥ 2 and min H > 0 on a closed surface S0 . Then under the MCF, the mean curvature
1
2
t)− 2 .
H(t) satisfies H ≥ Hmin (1 − n2 Hmin
Proof. We apply Theorem 61 of Appendix B.2, with f (u) = n1 u3 . The initial value problem ∂t ϕ = n1 ϕ3 and
1
2
ϕ|t=0 = min H|t=0 for ϕ is solved as ϕ(t) = Hmin (1 − n2 Hmin
t)− 2 , which implies the claimed relation.
18
Finally, we mention that the second relation of Theorem 18 implies ∂t |W |2 = ∆|W |2 − 2|∇W |2 + 2|W |4 .
As above, applying the maximum principle shows that
1
|W | ≤ Wmax (1 − CWmax t)−1 .
2
1
A collapsing solution is called of type I if |W | is bounded as |W | ≤ C(t∗ − t)− 2 . Otherwise, it is called
of type II (see Huisken). It was conjectured that the generic collapse is of type I. Indeed, all collapses
investigated in the papers above are of type I.
MCF of submanifolds. For hypersurfaces immersed in manifolds, the theory of collapse and neckpinching
are expected to go through with minimal modification. The reason for this is that they are local theories.
The volume preserving flow is affected by the geometry of ambient space in a more substantial way, see
Section 8.
4
Normal hessians.
Recall that the hessian of a functional E(ϕ) is defined as Hess E(ϕ) := d grad E(ϕ). Note that unlike the
Gˆ
ateaux derivative, d, the gradient grad and therefore the hessian, Hess, depends on the Riemann metric on
the space on which E(ϕ) is defined.
The fact that geometrical quantities are independent of the parameterization, e.g. Ha (ϕ ◦ α) = Ha (ϕ),
where Ha (ϕ) := H(ϕ) − ahϕ, ν(ϕ)i, and that the tangential variations lead to reparametrization of the
surface, have the following spectral consequence
Proposition 21. The full hessian, Hess Va (ϕ), of the modified volume functional Va (ϕ), has the eigenvalue
0 with the eigenfunctions which are tangential vector fields on S
Proof. Given a (tangential) vector field ξ on the surface S, denote by ϕs a family of deformations of ϕ
satisfying ϕ0 = ϕ and ∂s ϕs |s=0 = ξ. (Consequently, there is a family αs of diffeormorphisms of U , s.t.
ϕ ◦ αs = ϕs and, in particular, with α0 = 1 and ∂s ϕ ◦ αs |s=0 = ξ). Since H(ϕ) and hϕ, ν(ϕ)i are independent
of reparametrization, ϕs satisfies again the soliton equation, Ha (ϕs ) = 0. Differentiating the latter equation
w.r.to s at s = 0 and using that ∂s ϕ ◦ αs |s=0 = ξ, is a tangential vector field, we obtain
dHa (ϕ)ξ = 0,
(4.1)
which proves the proposition.
Since the tangential variations lead to reparametrization of the surface to the eigenvalue 0 of the hessian,
H, we concentrate, in what follows, on normal variations, η = f ν. (In physics terms, specifying normal
variations is called fixing the gauge.) We use the following notation for a linear operator, A, on normal
vector fields on S: AN f = A(f ν). For instance, HessN E(ϕ)f = Hess E(ϕ)(f ν) and
dN F (ϕ)f = dF (ϕ)(f ν).
(4.2)
19
We consider the hessian of the modified volume functional Va (ϕ), at a self-similar ϕ (i.e. H(ϕ) R= aϕN )
and in the normal direction (i.e. for normal variations, η = f ν) in the Riemann metric h(ξ, η) := S λ ξηρ.
In what follows, we call this hessian the normal hessian and denote it by HessN Va (ϕ). We want to describe
it spectrum.
Theorem 22. The hessian, HessN Va (ϕ), of the modified volume functional Va (ϕ), at a self-similar ϕ (i.e.
H(ϕ) = aϕ · ν), which is not a plane (i.e. ϕ ≡
6 0), and in the normal direction (i.e. for normal variations,
η = f ν), has
• the eigenvalue −2a with the eigenfunction hϕ, ν(ϕ)i,
• the eigenvalue −a with the eigenfunctions ν j (ϕ), j = 1, . . . , n + 1,
• the eigenvalue 0 of the multiplicity 21 n(n − 1), if ϕ 6= sphere or cylinder, and of the multiplicity n − 1,
if ϕ = cylinder, and 0 is not an eigenvalue, if ϕ = sphere; in the first case, the eigenfunctions are
hσj ϕ, ν(ϕ)i, j = 1, . . . , 21 n(n − 1), where σj are generators of the Lie algebra of SO(n + 1).
Before we proceed to the proof of this result, we mention the following important property of Va (ϕ): the
equation H(ϕ) − ahϕ, ν(ϕ)i = 0 breaks the scaling, translational and rotational symmetry. Indeed, using the
relations
H(λϕ) = λ−1 H(ϕ), ν(λϕ) = ν(ϕ), ∀λ ∈ R+ ,
H(ϕ + h) = H(ϕ), ν(ϕ + h) = ν(ϕ), ∀h ∈ R
(4.3)
n+1
H(gϕ) = H(ϕ), ν(gϕ) = gν(ϕ), ∀g ∈ O(n + 1),
,
(4.4)
(4.5)
using that g ∈ O(n + 1) are isometries in Rn+1 and using the notation Ha (ϕ) := H(ϕ) − ahϕ, ν(ϕ)i, we obtain
Hλ−2 a (λϕ) = λ−1 Ha (ϕ), ∀λ ∈ R+ ,
Ha (ϕ + h) + ahh, ν(ϕ)i = Ha (ϕ), ∀h ∈ R
Ha (gϕ) = Ha (ϕ), ∀g ∈ O(n + 1).
(4.6)
n+1
,
(4.7)
(4.8)
Proof of Theorem 22. If an immersion ϕ satisfies the soliton equation H(ϕ) = ahϕ, ν(ϕ)i, then by (4.6), we
have Hλ−2 a (λϕ) = 0 for any λ > 0. Differentiating this equation w.r.to λ at λ = 1, we obtain dHa (ϕ)ϕ =
−2ahϕ, ν(ϕ)i.
Now, choosing ξ to be equal to the tangential projection, ϕT , of ϕ, and subtracting the equation
(4.1) from the last equation, we find dHa (ϕ)hϕ, ν(ϕ)iν(ϕ) = −ahϕ, ν(ϕ)i. Since by the definition (4.2),
dHa (ϕ)f ν(ϕ) = dN Ha (ϕ)f and Hess Va (ϕ) = dHa (ϕ), this proves the first statement.
To prove the second statement, we observe that the soliton equation implies, by (4.7), that Ha (ϕ + sh) +
ashh, ν(ϕ)i = 0 and any constant vector field h. Differentiating this equation w.r.to s at s = 0, we obtain
dHa (ϕ)h = −ahh, ν(ϕ)i. Now, choosing ξ to be equal to the tangential projection, hT , of h, and subtracting
the equation (4.1) from the last equation, we find dHa (ϕ)hh, ν(ϕ)iν(ϕ) = −ahh, ν(ϕ)i, which together with
(4.2) gives the second statement.
Finally, to prove the third statement, we differentiate the equation Ha (g(s)ϕ) = 0, where g(s) is a oneparameter subgroup of O(n + 1), w.r.to s at s = 0, to obtain dHa (ϕ)σϕ = 0, where σ denotes the generator
20
of g(s). Now, choosing ξ in (4.1) to be equal to the tangential projection, (σϕ)T , of σϕ, and subtracting the
equation (4.1) from the last equation, we find dHa (ϕ)hσϕ, ν(ϕ)iν(ϕ) = 0, which together with (4.2) gives the
third statement. Finally, an explicit and elementary computation performed below shows that the normal
hessian, HessN
sph Va (ϕ) on a sphere has no eigenvalue 0.
In what follows a > 0.
Corollary 23. The multiplicity of the eigenvalue 0 (zero multiplicity meaning that 0 is not an eigenvalue)
differentiates between spheres, cylinders and other surfaces. In particular, a self-similar surface, ϕ, with
a > 0 satisfying HessN Va (ϕ) > 0 on the subspace span{hϕ, ν(ϕ)i, ν j (ϕ), j = 1, . . . , n + 1}⊥ , is a sphere.
Corollary 24. There are no smooth, embedded self-similar (a > 0) surfaces in Rn+1 , satisfying HessN Va (ϕ) ≥
0 on the subspace span{hϕ,q
ν(ϕ)i, ν j (ϕ), j = 1, . . . , n + 1}⊥ , which are close to S k × Rn−k , where S k is the
round k−sphere of radius
k
a
(for k = 1, . . . , n).
Theorem 22 implies that if ϕ is not spherically symmetric, then 0 is an eigenvalue of HessN Va (ϕ) of
multiplicity at least n + 1. This gives the second part of the first corollary, while the second one follows
from the fact that if there is a sequence of non-spherical self-similar surfaces approaching a sphere, then by
Theorem ?? and a standard spectral result, the normal hessian on the sphere would have the eigenvalue 0
which contradicts the third statement.
Remark 1. a) Strictly speaking, if the self-similar surface is not compact, then hϕ, ν(ϕ)i and ν j (ϕ), j =
1, . . . , n + 1, generalized eigenfunctions of HessN Va (ϕ). In the second case, the Schnol-Simon theorem (see
Appendix C.1 or [36]) implies that the points −2a and −a belong to the essential spectrum of HessN Va (ϕ).
pa
b) We show below that the normal hessian, HessN
sph Va (ϕ), on the sphere of the radius
n , given in
2a
(4.23), has no other eigenvalues below n , besides −2a and −a. A similar statement, but with n replaced by
n − 1, we have for the cylinder.
We call the eigenfunction hϕ, ν(ϕ)i, ν j (ϕ) and hσj ϕ, ν(ϕ)i, j = 1, . . . , n+1, the scaling, translational and
rotational modes. They originate from the normal projections of scaling, translation and rotation variations.
4.1
Spectra of hessians for spheres and cylinders
It is easy to show (for a systematic derivation see Subsection 4.5) that the normal hessians on the n−sphere
n−k
n
k
n
n
n
and (n,
q k)−cylinder, SR and the standard n−sphere S = S1 and the n−cylinder CR = SR × R of radius
R=
n−k
a
in Rn+1 , , are given by
a
HessN
sph Va (ϕ) = − ∆Sn − 2a,
n
(4.9)
on L2 (Sn ), where ∆Sk is the Laplace-Beltrami operator on the standard n−sphere Sk , and
HessN
cyl Va (ϕ) = −∆y − ay · ∇y −
a
∆ n−k − 2a,
n−k S
(4.10)
21
acting on L2 (C n ), where the standard round cylinder, C n = Sn−k × Rk , is naturally embedded in Rn+1 as
2
C n = {(ω, y) : |ω| = 1} ⊂ Rn+1 .
pa
To describe the spectra of the normal hessians on the n−sphere and (n, k)−cylinder, of the radii
n
q
a
and n−1
, we observe that it is a standard fact that the operator −∆ = −∆Sn is a self-adjoint operator on
L2 (Sn ). Its spectrum is well known (see [66]): it consists of the eigenvalues l(l + n − 1), l = 0, 1, . . . , of the
multiplicities
n+l
n+l−2
m` =
−
.
n
n
Moreover, the eigenfunctions corresponding to the eigenvalue l(l + n − 1) are the restrictions to the sphere
Sn of harmonic polynomials on Rn+1 of degree l and denoted by Ylm (the spherical harmonics),
− ∆Ylm = l(l + n − 1)Ylm ,
l = 0, 1, 2, 3, . . . , m = 1, 2, . . . , ml .
(4.11)
In particular, the first eigenvalue 0 has the only eigenfunction 1 and the second eigenvalue n has the eigenfunctions ω 1 , · · · , ω n+1 .
a
:= HessN
Consequently, the operator Lsph
a
sph Va (ϕ) = − n (∆Sn + 2n) is self-adjoint and its spectrum
a
consists of the eigenvalues n (l(l + n − 1) − 2n) = a(l − 2) + na l(l − 1), l = 0, 1, . . . , of the multiplicities
m` . In particular, the first n + 2 eigenvectors of Lsph
(those with l = 0, 1) correspond to the non-positive
a
eigenvalues,
0
0
j
j
Lsph
Lsph
(4.12)
a ω = −2aω ,
a ω = −aω , j = 1, . . . , n + 1,
and are due to the scaling (l = 0) and translation (l = 1) symmetries.
We proceed to the cylindrical hessian Lcyl
:= HessN
a
cyl Va (ϕ), given in (4.25). The variables in this
a
operator separate and we can analyze the operators −∆y − ay · ∇ and − n−k
(∆Sn−k + 2(n − k)) separately.
a
The operator − n−k (∆Sn−k + 2n) was already analyzed above. The operator −∆y − ay · ∇ is the Ornstein Uhlenbeck generator, which can be unitarily mapped by the gauge transformation
a
v(y, w) → v(y, w)e− 4 |y|
2
into the the harmonic oscillator Hamiltonian Hharm := −∆y + 14 a2 |y|2 − ka. Hence the linear operator
2
a
−∆y − ay · ∇ is self-adjoint on the Hilbert space L2 (R, e− 2 |y| dy). Since, as was already mentioned, the
a
operator − n−k (∆Sn−k + 2(n − k)) is self-adjoint on the Hilbert space L2 (C n ), we conclude that the linear
a
2
operator Lcyl
on the Hilbert space
L2 (Rk × Sn−k , e− 2 |y| dydw). Moreover, the spectrum of
a is self-adjoint
n P
o
k
−∆y − ay · ∇ is a 1 si : si = 0, 1, 2, 3, . . . , with the normalized eigenvectors denoted by φs,a (y), s =
(s1 , . . . , sk ),
(−∆y − ay · ∇)φs,a = a
k
X
si φs,a ,
si = 0, 1, 2, 3, . . . .
(4.13)
1
a
a
Using that we have shown that the spectrum of − n−k
(∆Sn−k +2(n−k)) is n−k
(l(l+n−k−1)−2(n−k)) =
P
k
a
1 si , si = 0, 1, 2, 3, . . ., we conclude the spectrum
n−k l(l + n − k − 1) − 2a, l = 0, 1, . . . , and denoting |s| =
of the linear operator Lcyl
,
for
k
=
1,
is
a
a
cyl
spec(La ) = (|s| − 2)a +
`(` + n − 2) : |s| = 0, 1, 2, 3, . . . ; ` = 0, 1, 2, . . . ,
(4.14)
n−1
22
with the normalized eigenvectors given by φs,l,m,a (y, w) := φs,a (y)Ylm (w). This equation shows that the
non-positive eigenvalues of the operator Lcyl
a , for k = 1 (so that |s| = s), are
1
a 4
• the eigenvalue −2a of the multiplicity 1 with the eigenfunction φ0,0,0,a (y) = ( 2π
) ((s, l) = (0, 0)), due
to scaling of the transverse sphere;
1
a 4 m
) w , m = 1, . . . , n
• the eigenvalue −a of the multiplicity n with the eigenfunctions φ0,1,m,a (y) = ( 2π
((s, l) = (1, 1)), due to transverse translations;
a 14 √
• the eigenvalue 0 of the multiplicity n with the eigenfunctions φ1,1,m,a (y) = ( 2π
) aywm , m = 1, . . . , n
((s, l) = (0, 1)), due to rotation of the cylinder;
a 41 √
) ay ((s, l) = (1, 0));
• the eigenvalue −a of the multiplicity 1 with the eigenfunction φ1,0,0,a (y) = ( 2π
1
a 4
) (1−ay 2 ) ((s, l) = (2, 0)).
• the eigenvalue 0 of the multiplicity 1 with the eigenfunction φ2,0,0,a (y) = ( 2π
The last two eigenvalues are not of the broken symmetry origin and are not covered by Theorem ??. They
indicate instability of the cylindrical collapse.
4.2
Connection of the spectrum and geometry of self-similar surfaces
For a 6= 0, the soliton equation, hϕ, ν(ϕ)i = a−1 H(ϕ), and Theorem 22 imply that the mean curvature H is
an eigenfunction of HessN Va (ϕ) with the eigenvalue −2a. This fact has important consequences, which are
used in the next result in order to connect the spectrum to the geometry of S.
Theorem 25. (Colding - Minicozzi) For a self-similar surface with a > 0, HessN Va (ϕ) ≥ −2a iff either
H(x) > 0 ∀x ∈ S, or H(x) < 0 ∀x ∈ S.
To prove this theorem we will use the Perron-Frobenius theory (see Appendix C.2) and its extension as
given in [60]. We begin with
Definition 1. We say that a linear operator on L2 (S) has a positivity improving property iff either A, or
e−A , or (A + µ)−1 , for some µ ∈ R, takes non-negative functions into positive ones.
Proposition 26. The normal hessian, HessN Va (ϕ), has a positivity improving property.
Proof. By standard elliptic/parabolic theory, e∆ and (−∆ + µ)−1 , for any µ > 0, has strictly positive integra
kernel and therefore is positivity improving. To lift this result to HessN Va (ϕ) := −∆ − |W |2 − a1 + ϕ · ∇
we proceed exactly as in [60].
Since, the operator HessN Va (ϕ) is bounded from below and has a positivity improving property, it
satisfies the assumptions of the Perron-Frobenius theory (see Appendix C.2) and its extension in [60] to the
case when the positive solution in question is not an eigenfunction. The latter theory, Proposition 1 and
Remark 1 imply the statement of Theorem 25.
23
Corollary 27. Let ϕ be a self-similar surface. We have
(a) For a < 0 (ϕ is an expander), H(ϕ) changes the sign.
(b) For a = 0, inf HessN Va (ϕ) < 0.
(c) For a > 0, if ϕ is an entire graph over Rn and inf HessN Va (ϕ) ≥ −2a, then ϕ is a hyperplane.
Indeed, (a) follows directly from Theorems 22 and 25, while (b) follows from the fact that for a = 0, 0 is
an eigenvalue of the multiplicity ≥ n + 2 and therefore, by the Perron-Frobenius theory, it is not the lowest
eigenvalue of HessN Va (ϕ). (c) is based on the fact that entire graphs over Rn cannot have strictly positive
mean curvature. (check, references)
Theorems 25 and 11 imply
Corollary 28 (Colding - Minicozzi). The only smooth, complete, embedded self-similar (a > 0) surfaces
in Rn+1 of polynomial
growth, satisfying HessN Va (ϕ) ≥ −2a, are S k × Rn−k , where S k is the round
q
k−sphere of radius
4.3
k
a.
Linear stability of self-similar surfaces
Given a self-similar surface ϕ, we consider for example the manifold of surfaces obtained from ϕ by symmetry
transformations,
Mself−sim := {λgϕ + z : (λ, z, g) ∈ R+ × Rn+1 × SO(n + 1)}.
By the spectral theorem above, it has unstable and central manifolds corresponding to the eigenvalues
−2a, −a and 0. Hence, we can expect only the dynamical stability in the transverse direction.
Definition 2 (Linearized stability of self-similar surfaces). We say that a self-similar surface φ, with a > 0, is
linearly stable (for the lack of a better term) iff HessN Va (ϕ) > 0 on the subspace span{hϕ, ν(ϕ)i, ν i (ϕ), i =
⊥
⊥
1, . . . , n+1, hσj ϕ, ν(ϕ)i, j = 1, . . . , 21 n(n−1)} (i.e. on span{scaling, translational, rotational modes} ).
(I.e. the only unstable motions allowed are scaling, translations and rotations.)
A weaker notion of stability was introduced in [17]:
Definition 3 (F −stability of self-similar surfaces of [17]). We say that a self-similar surface φ, with a > 0,
is F −stable iff the normal hessian satisfies HessN Va (ϕ) ≥ 0 on the subspace span{hϕ, ν(ϕ)i, ν j (ϕ), j =
1, . . . , n + 1}⊥ .
Remark 2. 1) The F −stability, at least in the compact case, says that the HessN Va (ϕ) has the smallest
possible negative subspace, i.e ϕ has the smallest possible Morse index.
2) The reason the F −stability works in the non-compact case is that, due to the separation of variables
for the cylinder = (compact surface) ×Rk , the orthogonality to the negative eigenfunctions of the compact
factor removes the entire branches of the essential spectrum. This might not work for warped cylinders.
24
3) For minimal surfaces (strict) stability implies the the surface is a (strict) minimizer of the volume
functional V (ψ). As is already suggested by the discussion above, this is not so for self-similar surfaces, they
are saddle points possibly satisfying some min-max principle.
4) Remark 1(c) shows that the spheres and cylinders are F −stable. However, we show in Section 6 that
cylinders are dynamically unstable.
N
cyl
By the description of the spectra of the normal hessians of Lsph
:= HessN
a
sph Va (ϕ) and La := Hesscyl Va (ϕ)
in the previous subsection, we conclude that the spherical collapse is linearly stable while the cylindrical one
is not.
We will show in Sections 5 and 6 that indeed the spherical collapse is (nonlinearly) stable while the
cylindrical one is not. We will also show that the last two eigenvalues of Lcyl
a in the list above are due to
translations of the point of the neckpinch on the axis of the cylinder and due to shape instability, respectively.
4.4
Explicit expression for the normal hessians
Now, we compute explicit expression for the normal hessian, HessN Va (ϕ).
Theorem 29. The normal hessian, HessN Va (ϕ) at a self-similar ϕ (i.e. H(ϕ) = aϕ · ν) is given by
HessN Va (ϕ) = −∆ − |W |2 − a(1 − ϕ · ∇).
(4.15)
We begin with
Theorem 30. The normal hessian, HessN V (ϕ), of the area functional V (ϕ) at a surface ϕ is given by
HessN V (ϕ) = −∆ − |W |2 .
(4.16)
Proof. Recall that Rby the definition (1.8) and the formula (2.9), we have gradh V (ϕ) = Hν in the Riemann
metric h(ξ, η) := S λ ξηρ. For normal variations, η = f ν, this gives gradN
h V (ϕ) = H(ϕ). We have to
compute d gradh V norm (ϕ) = dHν w.r.to normal variations. We claim that
dN Hf = (−∆ − |W |2 )f.
(4.17)
To prove this relation, we begin with some simple calculations. We have
Lemma 31 (Simons, Schoen-Simon-Yau, Hamilton, Huisken, Sesum). We have the following identities
(a) dν(ϕ)η = −g ij hν, ∂i ηi∂uj ϕ = −hν, ∇ηi.
(b) dbij (ϕ)η = −hν, (∂i ∂j + Γkij ∂k )ηi.
(c) dg ij η = g im g jn dgmn η, dgmn η = h∂m ϕ, ∂n ηi + h∂n ϕ, ∂m ηi.
Proof. Denote ϕj := ∂uj ϕ and ηj := ∂uj η. For (a), use that hν(ϕs ), ϕsi i = 0 to obtain h∂s ν(ϕs ), ϕsi i =
−hν(ϕs ), ∂s ϕsi i, which implies hdν(ϕ)η, ϕi i = −hν(ϕ), ηi i and therefore dν(ϕ)η = g ij hdν(ϕ)η, ϕi iϕj =
−g ij hν(ϕ), ηi iϕj , which gives the desired statement.
25
2
ϕ
To prove (b), we use bij = −h ∂u∂i ∂u
, νi (see Lemma 57 of Appendix A.2), relation (a) and the equation
j
2
2
2
2
ϕ
ϕ
dϕη = η, to find dbij (ϕ)η = −h ∂u∂i ∂uj dϕη, νi−h ∂u∂i ∂u
, dνηi = −h ∂u∂i ∂uj η, νi+g mn hν, ηm ih ∂u∂i ∂u
, ϕn i. Next,
j
j
using the Gauss formula
∂2ϕ
= bij ν + Γkij ϕk
(4.18)
∂ui ∂uj
2
(see e.g. [57]), hν, ϕn i = 0 and hϕk , ϕn i = gkn gives dbij (ϕ)η = −h ∂u∂i ∂uj η, νi − g mn gkn Γkij hν, ∂m ηi, which
implies the statement (b).
The first statement in (c) follows by differentiating the relation gg −1 = 1, where g = (gij ), which gives
d(g ) = g −1 dgg −1 , and the second, from the definition gij = hϕi , ϕj i and the relation dϕη = η.
−1
(Cf. (c) with (1.12) (dgij η = 2bij (ν · η)).)
Using the definition H = g ij bij , we find dHη = bij dg ij + g ij dbij . We start with the second term. Notice
that the definition ∆f = div grad f = ∇i ∇i f and relations to
div V = ∇i V i =
∂V i
i
i
ij ∂f
+ Γm
,
mi V , (grad(f )) = g
i
∂u
∂uj
(4.19)
∂
m
ij ∂f
ij
k
imply ∆f = ( ∂u
i + Γmi )g
∂uj . Now, take η = f ν. Then the last relation gives g (∂i ∂j + Γij ∂k )η =
k
ν∆f + f (∂i ∂j + Γij ∂k )ν + 2∂i f ∂i ν. Using this relation, the statement (c) of Lemma 31 and the relation
hν, ∂j νi = 0, we obtain
dbij η = hν, (∂i ∂j + Γkij ∂k )(f ν)i = ∆f + f hν, ∂i ∂j νi.
Furthermore, using hν, ∂j νi = 0, again, and using hξ, ηi = g mn hξ, ϕm ihϕn , ηi, we find hν, ∂i ∂j νi = −h∂i ν, ∂j νi =
−g mn h∂i ν, ϕm ihϕn , ∂j νi. Next, the relation h∂k ν, ϕl i = −hν, ∂k ϕl i = bkl gives hν, ∂i ∂j νi = −g mn bim bjn .
Collecting the above equalities, we arrive at
dbij (f ν) = ∆f − g mn bim bjn f,
(4.20)
and therefore g ij dbij (f ν) = ∆f − g ij g mn bim bjn f = ∆f − |W |2 f .
Next, for the first term in dH = bij dg ij + g ij dbij , we have, by Lemma 31 (c), dg ij = g im g jn dgmn . Now,
using Lemma 31 (c) again and using the relation hν, ∂j νi = 0, we find, for η = f ν,
dgmn (f ν) = hϕm , ∂n (f ν)i + hϕn , ∂m (f ν)i = f [hϕm , ∂n νi + hϕn , ∂m νi] = 2f bmn .
Combining the last two relations gives bij dg ij (f ν) = 2f g im g jn bmn bij = 2|W |2 f . Collecting the relations
above gives (4.17).
(Express (b) in terms of the connections, whose properties are given in Appendix A.4.)
Proof of Theorem 29. Recall that by the definition R(1.8) and the formula (2.9), we have gradh Va (ϕ) =
(H − ahϕ, νi)ν in the Riemann metric h(ξ, η) := S λ ξηρ. For normal variations, η = f ν, this gives
gradh Vanorm (ϕ) = (H(ϕ) − aϕ · ν). We have to compute d gradh Vanorm (ϕ) = d(H − ahϕ, νi)ν w.r.to normal
variations. We already computed dH and therefore it remains to compute dhϕ, νi. We claim that
dN hϕ, νif = f − hϕ, ∇if.
(4.21)
26
Using Lemma 31 and the equation dϕη = η, we find
dhϕ, νiη = hη, νi − g ij hν, ∂i ηihϕ, ϕj i.
(4.22)
Now, taking η = f ν and using ∂i (f ν) = ∂i f ν + f ∂i ν, hν, νi = 1 and hν, ∂j νi = 0, we arrive at (4.21).
4.5
Hessians for spheres and cylinders
We finish this section with illustrating the above result by computing the normal hessians on the n−sphere
and (n, k)−cylinder.
pn
n
n+1
Explicit expressions. 1) For the n−sphere SR
of radius R =
, we have bij = R−1 gij =
a in R
n
stand
stand
n
Rgij , where gij and gij
are the metrics on SR and the standard n−sphere S = S1n , respectively, which
2
−2
gives |W | = nR = a. Moreover, since ν(ϕ) = ϕ, we have that ϕ · ∇ = 0 and therefore
a
HessN
sph Va (ϕ) = − ∆Sn − 2a,
n
(4.23)
on L2 (Sn ), where ∆Sk is the Laplace-Beltrami operator on the standard n−sphere Sk .
q
n−k
n
n+1
stand
= SR
× Rk of radius R = n−k
2) For the n−cylinder CR
, we have bij = R−1 gij = Rgij
,
a in R
n−1
stand
for i, j = 1, . . . , n − k, and bαβ = δαβ , α, β = n − k + 1, . . . , n. Here gij and gij
are the metrics on SR
n−1
and Sn−1 = S1 , respectively. This gives
|W |2 = (n − k)R−2 = a.
(4.24)
Moreover, letting the standard round cylinder C n = S n−k × Rk be naturally embedded as C n = {(ω, x) :
2
|ω| = 1} ⊂ Rn+1 , we see that the immersion ϕ is given by ϕ(ω, y) = (χ(ω, y), y) for some χ(ω, y), where
y = y(x) := λ−1 (t)x. This gives ϕ · ∇ = ω · ∇Sn−k + y · ∇y = y · ∇y , which together with the previous relation,
implies
HessN
cyl Va (ϕ) = −∆y − ay · ∇y −
a
∆ n−k − 2a,
n−k S
(4.25)
acting on L2 (C n ).
4.6
Minimal and self-similar submanifolds
For a self-similar hypersurface S in a manifold M , with immersion ϕ (satisfying H(ϕ) = aϕ · ν), the normal
hessian, HessN Va (ϕ) is given by
HessN Va (ϕ) = −∆ − |W |2 − a(1 − ϕ · ∇) − RicM (ν, ν),
where RicM (ν, ν) is the Ricci curvature of M .
(4.26)
4.7
27
Translational solitons
Recall that the translation solitons are solutions of the MCF of the form S(t) ≡ S + h(t) (traveling waves),
or ψ(u, t) = ϕ(u) + z(t), where h(t) ∈ Rn+1 , with the profile function ϕ and the velocity v satisfying the
equation
H(ϕ) = −hν(ϕ), vi,
(4.27)
where v = z˙ is a constant vector. The translation solitons, (ϕ, v), are static solutions of the equation
(∂t ϕ)N = −Hv (ϕ),
(4.28)
where weR used the notation Hv (ϕ) := H(ϕ) + v N (ϕ) and v N (ϕ) := hv, ν(ϕ)i. Let ρv (x) = ev·x and
Vv (ϕ) := S λ ρv (conformal area of S λ ).
Proposition 32. For a surface S given locally by an immersion ψ and normal variations, η = f ν, we have
Z
dVv (ϕ)η =
Hv (ϕ)f ρv (ϕ)
(4.29)
S
p
R
Proof. The definition of Vv (ϕ) gives Vv (ϕ) = U ρh (ϕ) g(ϕ)dn u. For ρv (ϕ) = e−v·x , we have dρv (ϕ)η =
√
√
ρv (ϕ)hv, ηi. Using this formula and the equation d gη = H ghν, ηi proven above (see (1.10)) and the fact
that we are dealing with normal variations, η = f ν, we obtain (4.29).
By the definition (1.8), the definition of the normal gradient and the formula (2.9), we have
gradN
h Vv (ϕ) = Hv (ϕ)ν
R
in the Riemann metric h(ξ, η) := S λ ξ · ηρv , where ξ · η = hξ, ηi is the Euclidean inner product in Rn+1 .
This and the equation (4.28) and Proposition 32 imply
Corollary 33.
a) Assume v in (4.28) is constant. Then the modified area functional Vv (ϕ) is monotonically decaying under the flow (4.28), more precisely,
Z
∂t Vv (ϕ) = − ρv |H + v N |2 .
(4.30)
S
b) The renormalized Rflow (4.28) is a gradient flow for the modified area functional Vv (ϕ) and the Riemann
metric h(ξ, η) := S λ ξ · ηρv :
ϕ˙ = − gradh Vv (ϕ).
We consider the spectrum of the normal hessian on a translational soliton (ϕ, v). We use the notation
v N := hv, ν(ϕ)i, and (σij ϕ)N := hσj ϕ, ν(ϕ)i, where σj are generators of the Lie algebra of SO(n + 1).
Theorem 34. The normal hessian, HessN Vv (ϕ), of the volume functional Vv (ϕ), at a translation soliton
ϕ, which is not a plane (i.e. ϕ 6≡ 0), has the eigenvalue 0 of
• the multiplicity n + 1, with the eigenfunctions ν j (ϕ), j = 1, . . . , n + 1, if ϕ is not spherical or axisymmetric;
28
• the multiplicity n, with the eigenfunctions ν j (ϕ), j = 1, . . . , n, if ϕ is axi-symmetric;
• the multiplicity n + 1, with the eigenfunctions ν j (ϕ), j = 1, . . . , n + 1, if ϕ is a sphere.
Proof. As before, we use the fact that the equation (4.27) and its solution breaks the scaling, translational
and rotational symmetry. Differentiating the relations (4.3) - (4.5), w.r. to the corresponding parameters
and using (4.27), we obtain
dHv (ϕ)ϕ = v N , dHv (ϕ)z = 0, dHv (ϕ)σj ϕ = (σj∗ v)N .
(4.31)
where, recall, the notation Hv (ϕ) := H(ϕ) + v N . Now, removing the tangential components of ϕ, z and
σij ϕ gives
dHv (ϕ)ϕN = v N ,
dHv (ϕ)z
N
(4.32)
= 0,
N
dHv (ϕ)(σj ϕ)
(4.33)
=
(σj∗ v)N .
(4.34)
The second equation implies the statement of the theorem.
Definition 4 (Linearized stability of the translation solitons). We say that a translation soliton, (ϕ, v), is
linearly stable (for the lack of a better term) iff HessN Vv (ϕ) ≥ 0.
5
Stability of elementary solutions I. Spherical collapse
The next two results indicate some possible scenarios for the long-time behavior of surface under mean
curvature flow.
Theorem 35.
• (Ecker-Huisken) The hyperplanes are asymptotically stable.
• (Kong-Sigal) The spherical collapse is asymptotically stable.
• (Gang - Sigal) The cylindrical collapse is unstable.
We prove the second statement which we formulate precisely. Recall that we are considering the mean
curvature flow, starting with a hypersurface S0 in Rn+1 , which is given by an immersion, ψ0 : Sˆ → Rn+1 ),
of some fixed n−dimensional hypersurface Sˆ ⊂ Rn+1 (i.e. ψ(·, t) : Sˆ → Rn+1 ). We look for solutions of the
mean curvature flow,
∂ψ
= −H(ψ)ν(ψ)
∂t
(5.1)
ψ|t=0 = ψ0 ,
as immersions, ψ(·, t) : Sˆ → S, of Sˆ giving the hypersurfaces S(t). (Recall that H(x) and ν(x) are mean
curvature and the outward unit normal vector, at x ∈ S(t), respectively.)
Theorem 36. Let a surface S0 , defined by an immersion x0 ∈ H s (Sn ), for some s > n2 + 1, be close to
Sn , in the sense that kψ0 − 1kH s 1. Then there exist t∗ < ∞ and z∗ ∈ Rn+1 , s.t. (5.1) has the unique
29
solution, S(t), t < t∗ , and this solution contracts to the point z∗ , as t∗ → ∞. Moreover, S(t) is defined, up
to a (time-dependent) reparametrization, by an immersion ψ(·, t) ∈ H s (Sn ), with the same s, of the form
ψ(ω, t) = z(t) + λ(t)ρ(ω, t)ω,
(5.2)
where ρ, λ(t), z(t) and a(t) satisfy
√
2a∗ s + O(sκ1 ),
˙
a(t) = −λ(t)λ(t)
= a∗ + O(sκ2 ),
(5.3)
κ3
(5.5)
λ(t) =
(5.4)
z(t) = z∗ + O(s )
r
1
n
kρ(·, t) −
kH s . s 2n ,
a(t)
with s := t∗ − t, κ1 :=
1
2
+
1
2a∗ (1
−
1
2n ),
κ2 :=
1
2a∗ (1
−
1
2n )
and κ3 :=
(5.6)
1
2a∗ (n
+
1
2
−
1
2n ).
Moreover, |z∗ | 1.
Note that our condition on the initial surface does not impose any restrictions of the mean curvature of
this surface.
The form of expression (5.2) above is a reflection of a large class of symmetries of the mean curvature
flow:
• (5.1) is invariant under rigid motions of the surface, i.e. x 7→ Rx + a, where R ∈ O(n + 1), a ∈ Rn+1
and x = x(u, t) is a parametrization of S(t), is a symmetry of (5.1).
• (5.1) is invariant under the scaling x 7→ λx and t 7→ λ−2 t for any λ > 0.
We utilize these symmetries in an essential way by defining the centres, z(t), of the closed surfaces S(t) and
using the rescaling of the equation (5.1) by a parameter λ(t). This leads to a transformed MCF depending
explicitly on these parameters and determining their behaviour. Then we define Lyapunov-type functionals
and derive a series of differential inequalities for them from which we obtain our main results. We omit some
technical details for which we refer to [56, 3].
Notation. The relation f . g for positive functions f and g signifies that there is a numerical constant
C, s.t. f ≤ Cg. In this section we denote the inner product in Rn+1 by the dot: x · y.
5.1
Rescaled equation
˜ ) = λ−1 (t)(S(t) − z(t)),
Instead of the surface S(t), it is convenient to consider the new, rescaled surface S(τ
Rt
where λ(t) and z(t) are some differentiable functions to be determined later, and τ = 0 λ−2 (s)ds. The new
surface is described by ϕ, which is, say, an immersion of some fixed n−dimensional hypersurface Ω ⊂ Rn+1 ,
˜ ), i.e. ϕ(·, τ ) : U → S(t)). Thus the new collapse
i.e. ϕ(·, τ ) : Ω → Rn+1 , (or a local parametrization of S(τ
variables are given by
Z
t
ϕ(ω, τ ) = λ−1 (t)(ψ(ω, t) − z(t)) and τ =
λ−2 (s)ds.
0
(5.7)
30
R t −2
∂z
Let λ˙ = ∂λ
(s)ds.
∂t and ∂τ be the τ -derivative of z(t(τ )), where t(τ ) is the inverse function of τ (t) = 0 λ
∂ϕ ∂τ
∂ψ
∂z
−2 ∂z
−1 ∂ϕ
−1
˙
˙
Using that ∂t = ∂t + λϕ + λ ∂τ ∂t = λ ∂τ + λϕ + λ ∂τ and H(λϕ) = λ H(ϕ), we obtain from (5.1)
the equation for ϕ, λ and z:
∂ϕ
∂z
˙
· ν(ϕ) = −H(ϕ) + (aϕ − λ−1 ) · ν(ϕ) and a = −λλ.
∂τ
∂τ
(5.8)
Thus S(t) is given by the datum (ϕ, λ, z), satisfying
p Eq. (5.8). Note that the equation (5.8) has static
solutions (a = a positive constant, z = 0, ϕ(ω) = na ω, ω ∈ Sn ).
5.2
The graph represetation
In the next proposition ∆ is the Laplace-Beltrami operator in is the standard metric on Sn , ∇i ρ =
2
∂ ρ
∂ui uj
∂ρ
− Γkij ∂u
k,
Γkij
1 kn ∂gin
2 g ( ∂uj
∂gjn
∂ui
∂ρ
∂ui
(in a
∂gij
∂un ).
n
+
−
Here
where
=
local parametrization x = x(u)) and (Hess ρ)ij =
and in what follows the summation over the repeated indices is assumed. (Hess is the hessian on S and, if ∇
∂v
the Riemann connection on Sn , which acts on vector fields as (∇i v)j = ∂uji − Γkij vk , then (Hess)ij = ∇i ∇j .)
For more details see Appendix A. In this section we present the following
˜ ) = λ−1 (t)(S(t) − z(t)), for 0 ≤ t < T , for some T > 0, and τ , given in (5.7), and
Proposition 37. Let S(τ
˜ ) is defined by an immersion
for λ(t) and z(t) differentiable functions on [0, T ), and assume S(τ
ϕ(ω, τ ) = ρ(ω, τ )ω
(5.9)
˜ ) satisfies (5.8) if
of Sn for some functions ρ(·, τ ) : Sn → R+ , differentiable in their arguments. Then S(τ
and only if ρ and z satisfy the equation
∂ρ
= G(ρ) + aρ − λ−1 zτ · ω + λ−1 z˜τ · ρ−1 ∇ρ,
∂τ
where z˜τ k =
∂xi i
z ,
∂uk τ
zτ :=
∂z
∂τ
(5.10)
and (check the sign of the last term on the r.h.s.)
G(ρ) =
1
n ∇ρ · Hess(ρ)∇ρ − ρ|∇ρ|2
.
∆ρ − −
2
ρ
ρ
ρ2 (ρ2 + |∇ρ|2 )
(5.11)
Proof. By a reparametrization (see Proposition 1), the equation (5.8) is equivalent to the equation
∂ϕ
∂z
· ν(ϕ) = −H(ϕ) + (aϕ − λ−1 ) · ν(ϕ).
∂τ
∂τ
(5.12)
∂ρ
−1/2
For the graph ϕ(ω, τ ) = ρ(ω, τ )ω, we have ∂ϕ
ρ and
∂τ = ∂τ ω. By results of Appendix A.8, we have ν ·ω = p
−1/2 2
2
2
hν, ϕi = p
ρ , where p := ρ + |∇ρ| . These relations, together with the equation (5.12), the expression
∂z
(A.34) for the mean curvature H(ϕ) and a similar computation ∂τ
· ν(ϕ) = zτ · ω − +˜
zτ · ρ−1 ∇ρ, give the
equation (5.10).
We consider (5.10) as an equation for the unknowns ρ(t), z(t) and a(τ ), with λ(t) determined from the
˙
equation λ(t)λ(t)
= −a(τ (t)) (see (5.8)). Additional equations come from imposing the subsidiary conditions
r
n
,
(5.13)
hρ(·, τ )iSn =
a(τ )
31
R
where by hf iSn we denote the average, hf iSn := |S1n | Sn f , of f over Sn , fixing the average radius of the
rescaled surface, and
Z
ρ(ω, τ )ω j = 0, j = 1, . . . , n + 1,
(5.14)
Sn
fixing the center ofR the rescaled surface. To see the latter we observe that the latter conditions are equivalent
to the conditions Sn ((ψ − z) · ω)ω j = 0, j = 1, . . . , n + 1, which can be said to determine the center of the
surface.
5.3
Reparametrization of solutions
Our goal is to decompose a solution, ρ(ω, τ ), of the equation (5.10) into a leading part which is a sphere of
some radius τ -dependent and small fluctuation around this part. This would give a convenient reparametrization of our solution.
We decompose ρ(ω, τ ) into the sphere, which (a) is closest to it and (b) is a stationary solution to (5.10),
and a remainder. By (a)
q the remainder (fluctuation) is orthogonal to this sphere and by (b), the radius of
n
the sphere is equal to a(τ
) , where a(τ ) is the parameter-function appearing in (5.10). In other words, we
write
r
ρ(·, τ ) =
n
+ φ(ω, τ ),
a(τ )
with φ ⊥ 1 in L2 (Sn ).
(5.15)
Introduce the notation ω 0 = 1. Then the relations (5.15), (5.13) and (5.14) give
φ(·, τ ) ⊥ ω j , j = 0, . . . , n + 1, in L2 (Sn ).
5.4
(5.16)
Lyapunov-Schmidt decomposition
(changed φ to ξ) Let ρ solve (5.10) and assume it can be written as ρ(ω, τ ) = ρa(τ ) +ξ(ω, τ ), with ρa =
and ξ ⊥ ω j , j = 0, . . . , n + 1. Plugging this into equation (5.10), we obtain the equation
∂ξ
= −La ξ + Nλ,z (ξ) + F,
∂τ
pn
a
(5.17)
where La = −dGa (ρa ), with Ga (ρ) := G(ρ) + aρ, ρ = ρa + ξ, N (ξ) = G(ρa + ξ) − G(ρa ) − dG(ρa )ξ,
∂z
∂a
∂xi i
F = −∂τ ρa − λ−1 zτ · ω and, recall, zτ = ∂τ
and z˜τ k = ∂u
k zτ . Let aτ = ∂τ . We compute
√
n −3/2
a
La = (−∆ − 2n), F =
a
aτ − λ−1 zτ · ω,
(5.18)
n
2
Nλ,z (ξ) := N (ξ) + λ−1 z˜τ · ρ−1 ∇ξ,
(5.19)
N (ξ) = −
(ρa + ρ)ξ∆ξ
nξ 2
∇ξ · Hess(ξ)∇ξ − ρ|∇ξ|2
− 2 −
.
2
2
ρ ρa
ρρa
ρ2 (ρ2 + |∇ξ|2 )
(5.20)
Now, we project (5.17) onto span{ω j , j = 0, . . . , n + 1}. In the rest of the section, all the norms and
inner products are in the sense of L2 (Sn ). By ξ⊥ω j , j = 0, . . . , n + 1, we have
√
n −3/2
a
aτ |Sn | = hNλ,z (ξ), 1i ,
(5.21)
2
32
− cλ−1 zτj = Nλ,z (ξ), ω j ,
where j = 1, . . . , n + 1, and c :=
R
Sn
(ω j )2 =
1
n
n+1 |S |.
(5.22)
Indeed, this equation follows from
(i) ∂τ ξ, ω j = − ξ, ∂τ ω j = 0, La ξ, ω j = ξ, La ω j = 0, j = 0, . . . , n + 1;
D√
E √
(ii) hF, 1i = 2n a−3/2 aτ , 1 = 2n a−3/2 aτ |Sn |; F, ω j = −λ−1 zτ · ω, ω j = −cλ−1 zτj ,
j = 1, . . . , n + 1.
Projecting (5.17) onto the orthogonal complement of span{ω j , j = 0, . . . , n + 1}, we find
∂ξ
⊥
= −La ξ + Nλ,z
(ξ),
∂τ
(5.23)
⊥
˙
where Nλ,z
(ξ) is the corresponding projection of Nλ,z (ξ). We use the equations (5.21) - (5.23) and λ(t)λ(t)
=
−a(τ (t)) in order to determine a(τ ), z(t), ξ(t) and λ(t).
Remark 3. The operator La = −dGa (ρa ) is the hessian Hess Va (ρa ) = d grad Va (ρa ) of the modified volume
functional Va (ρ) at the sphere ρa .
5.5
Linearized operator
p
In this section we discuss the operator La := −∂Ga ( na ) = na (−∆ − 2n) acting on L2 (Sn ), which is the
p
linearization of the map −Ga (ρ) = −G(ρ) − aρ at ρa = na . We have already encountered this operator in
Section 2, as the normal hessian of modified volume functional around a sphere. Its spectrum was described
in that section. Here we mention only the following relevant for us fact: The only non-positive eigenvalues
of La are
La ω 0 = −2aω 0 , La ω j = −aω j , j = 1, . . . , n + 1
(5.24)
and the next eigenvalue is
2a
n
and consequently, on their orthogonal complement we have
hξ, La ξi ≥
2a
kξk2 if ξ ⊥ ω j , j = 0, . . . , n + 1.
n
(5.25)
This coercivity estimate will play an important role in our analysis. This is the reason why we need the
conditions (5.16).
The fact that the n + 2 non-positive eigenvectors of La are related to change of the radius and position
of the euclidean sphere comes from the scaling and translational symmetries of (5.1). Indeed, let α = (r, z),
where r := na , and ρα be the real function on Sn , whose graph is the sphere Sα in Rn+1 of radius R, centered
at z ∈ Rn+1 , i.e. graph(ρ) := {ρ(ω)ω : ω ∈ S n } = Sα . The function ρα satisfies the equation |ρα (x)ˆ
x −z| = r,
ˆ = r2 , and therefore it is given by
or ρα (x)2 + |z|2 − 2ρα (x)z · x
p
ρα (ˆ
x) = z · x
ˆ + r2 − |z|2 + (z · x
ˆ )2 ,
where, recall, x
ˆ=
x
|x| .
By the symmetries, the equation Ga (ρα ) = 0 holds for any α. Differentiating it w.r.to α, we find
∂Ga (ρα )∂α ρα + ∂α Ga (ρα ) = 0. If we introduce the operator Lα := −∂Ga (ρα ), then the latter equations
33
become Lα ∂r ρα = −2a∂r ρα , Lα ∂z ρα = 0, i. e. ∂α ρα are eigenfunctions of the operator Lα .
Lα = La + O(|z|) and
ˆj + O(|z|),
∂r ρα (x) = 1 + O(|z|), ∂zj ρα (x) = x
Since
(5.26)
which gives - in the leading order - the n + 2 non-positive eigenvectors of La , these equations imply (5.24).
The n + 2 non-positive eigenvectors of La span the unstable - central subspace of the tangent space of
the fixed point manifold, {Sr,z |r ∈ R+ , z ∈ Rn+1 } (manifold of spheres), for the rescaled MCF. We use
the parameters of size, a, of thep
sphere (1 parameter) and its translations z (n + 1 parameters) to make the
fluctuation ξ(ω, τ ) := ρ(ω, τ ) − na orthogonal to the unstable-central modes, so that to be able to control
it.
5.6
Lyapunov functional
We introducep
the operator L = −∆ − 2n, acting on L2 (Sn ), which is related to the linearized operator
La := −∂Ga ( na ) = na (−∆ − 2n), introduced above, as La := na L. Using the spectral information about
obtained above we see that on functions ξ obeying (5.17) and ξ⊥ω j , j = 0, . . . , n + 1, tit is bounded below
as hL0 ξ, ξi ≥ kξk2 , we derive in this section some differential inequalities for certain Sobolev norms of such
a ξ. These inequalities allow
us to prove a priori estimates for these Sobolev norms. For k ≥ 1, we define
the functional Λk (ξ) = 21 ξ, Lk ξ .
Using the coercivity estimate (5.25) and standard elliptic estimates, we obtain (see Proposition 8 of [3],
with R2 = n/a)
Proposition 38. There exist constants c > 0 and C > 0 such that
ckξk2H k ≤ Λk (ξ) ≤ Ckξk2H k .
Proposition 39. Let k >
n
2
+ 1. If φ satisfies (5.17), then there exists a constant C > 0 such that
∂τ Λk (ξ) ≤ −
k+1
a
2a
Λk (ξ) − [
− C(Λk (ξ)1/2 + Λk (ξ)k )]kL 2 ξk2 .
n
2n
(5.27)
Proof. We have, using (5.17), or (5.23),
a
1
ξ, Lk ξ = ∂τ ξ, Lk ξ = − Lξ, Lk ξ + Nλ,z (ξ), Lk ξ ,
(5.28)
2
n
D k
E
k+1
k
where, recall, we use the notation ρ = ρa + ξ. We write Lξ, Lk ξ = 21 kL 2 ξk2 + 21 L 2 ξ, LL 2 ξ and use
D k
E D k
E
k
k
the coercivity estimate (5.25) for the second term, L 2 ξ, LL 2 ξ ≥ L 2 ξ, L 2 ξ = Λk (ξ), to obtain
∂τ
1 k+1
Lξ, Lk ξ ≥ kL 2 ξk2 + 2Λk (ξ).
2
(5.29)
Now, we begin estimating the next term
D k−1
E
k+1
| Nλ,z (ξ), Lk ξ | = | La 2 Nλ,z (ξ), L 2 ξ |
≤ kL
k−1
2
Nλ,z (ξ)kkL
k+1
2
ξk.
(5.30)
34
Recall the definition (5.19) of Nλ,z (ξ). To estimate the contribution of the first term on the r.h.s. in (5.19),
we use the inequality, which is a special case of Lemma 12 of Appendix D of [3],
kL
k−1
2
1/2
N (ξ)k . (Λk (ξ) + Λkk (ξ))kL
k+1
2
ξk.
(5.31)
(The operator La of this paper
is identified with the first (main) part of the operator LR0 := − R12 (∆ + n) +
R
1
n
2
R2 Av, where Av ξ := |Γ| Γ ξ, of [3], once we set R = n/a.) (To prove estimates of the type of (5.31), one
uses the Leibnitz rule for fractional derivatives and fractional integration by parts in the Besov spaces, in
k−1
order to distribute the derivatives in L 2 N (ξ) evenly among various factors, see [3].)
To estimate the contribution of the second term on the r.h.s. in (5.19), we use Eq. (5.22) to obtain
λ−1 zτ . N (ξ) + λ−1 z˜τ · ρ−1 ∇ξ, ω . kN (ξ)kL1 + λ−1 |zτ |k∇ξkL1 .
Next, using (5.20), where, recall, ρ = ρa + ξ and ρa =
pn
a,
(5.32)
and assuming that |ξ| ≤ 21 ρa , we have that
kN (ξ)kL1 . (k∇ξk2L4 + kξkH 1 )kξkH 2 .
(5.33)
This together with (5.32) gives, provided that kξkH 1 1,
|zτ | . λ(k∇ξk2L4 + kξkH 1 )kξkH 2 .
(5.34)
The last two equations, together with Proposition 38, imply that
kL
k−1
2
1/2
(λ−1 z˜τ · ρ−1 ∇ξ)k ≤ C(Λk (ξ) + Λkk (ξ))kL
k+1
2
ξk.
(5.35)
Collecting the estimates (5.30), (5.31) and (5.35), we find
k+1
1/2
| Nλ,z (ξ), Lk ξ | ≤ C(Λk (ξ) + Λkk (ξ))kL 2 ξk2 .
(5.36)
Relations (5.28), (5.29) and (5.36) yield (5.27).
Remark 4. (5.27) is an approximate monotonicity formula for the Lyapunov functional Λk (ξ). In general,
the functional Λ(ξ) = na Λ1 (ξ) = 21 hξ, La ξi is the leading term in the expansion of the modified volume
functional Va (ϕ + ξ) at the self-similar surface ϕ. What is the relation between the monotonicity formula
(2.10) and the inequality (5.27)?
5.7
Proof of Theorem 36
We begin with reparametrizing the initial condition. First, we write the immersion ψ0 (ω) as ψ0 (ω) =
z0 + ϕ0 (ω), where z0 ∈ Rn+1 is s.t.
Z
(ψ0 (ω) − z0 ) · ω)ω j = 0, j = 1, . . . , n + 1.
(5.37)
Sn
so that
R
Sn
(ϕ0 (ω) · ω)ω j = 0.
35
Next, reparametrizing
Sn , if necessary, we can assume that ϕ0 (ω) is of the form ϕ0 (ω) = ρ0 (ω)ω, which
R
j
implies that Sn ρ0 (ω)ω = 0, j = 1, . . . , n + 1. Given ρ0 we define a0 by an0 = hρ0 iSn , where, recall,
R
hf iSn := |S1n | Sn f , the average of f over Sn , so that
r
ρ0 (ω) =
n
+ ξ0 (ω),
a0
with ξ0 ⊥ 1 in L2 (Sn ).
(5.38)
The last two statements imply that ξ0 ⊥ ω j , j = 0, . . . , n + 1, where, recall, ω 0 = 1.
1
Since the initial condition, ψ0 (ω), is sufficiently close to the identity, ξ0 (ω) and a0 satisfy Λk (ξ0 ) 2 +
1
1
, Λk (ξ0 ) 1 and |a0 − n| ≤ 10
(see (5.38)), where the constant C is the same as in
Λk (ξ0 )k ≤ 10C
Proposition 39.
˙
We consider the equations (5.21) - (5.23) and λ(t)λ(t)
= −a(τ (t)) for a(τ ), z(t), ξ(t) and λ(t). The
˙
initial value problem λ(t)λ(t) = −a(τ (t)) for λ(t) and λ(t0 ) = λ0 , where τ = τ (t) given by (5.7), can be
solved explicitly to obtain λ(t) = λ(a, λ0 )(t), where λ(a, λ0 )(t) is the positive function given by
Z t
2
λ(a, λ0 )(t) := (λ0 − 2
a(τ (s))ds)1/2 .
(5.39)
t0
To solve for a(τ ), z(t) and ξ, we use the spaces At0 ,δ , Zt0 ,δ = C 1 (It0 ,δ , Rn+1 ) and H s (Sn ), s > n/2 + 1,
respectively. Here, t0 ≥ 0, δ > 0, It0 ,δ := [t0 , t0 + δ], and
1
1
At0 ,δ := C 1 (It0 ,δ , [n − , n + ]).
2
2
(5.40)
In what follows we deal only with the unknowns a(τ ), z(t) and ξ(t) and consider λ(t) as given by (5.39).
Now, a standard ODE and parabolic theory gives the local existence result for the equations (5.21) - (5.23).
Now, let
a
}.
4nC
By continuity, T > 0. Assume T < ∞. Then ∀τ ≤ T we have by Proposition 39 that ∂τ Λk (ξ) ≤ − 2a
n Λk (ξ).
1
We integrate this equation to obtain Λk (ξ) ≤ Λk (ξ0 )e− 2 τ ≤ Λk (ξ0 ). This implies
1
T = sup{τ > 0 : Λk (ξ(τ )) 2 + Λk (ξ(τ ))k ≤
1
1
Λk (ξ(T )) 2 + Λk (ξ(T ))k ≤ Λk (ξ0 ) 2 + Λk (ξ0 )k ≤
a
.
8nC
1
a
Therefore proceeding as above we see that there exists δ > 0 such that Λk (ξ(τ )) 2 + Λk (ξ(τ ))k ≤ 4nC
, for
τ ≤ T + δ, which contradicts
the assumption that T < ∞ is the maximal existence
time. So T = ∞ and
R
R
τ
τ
2
2
Λk (ξ) ≤ Λk (ξ0 )e− n 0 a(s)ds . By Proposition 38 we know that kξk2H k . Λk (ξ0 )e− n 0 a(s)ds .
Now, we use Eq. (5.21) to obtain
−3/2 a
aτ . N (ξ) + λ−1 z˜τ · ρ−1 ∇ξ, 1 . kN (ξ)kL1 + λ−1 |zτ |k∇ξkL1
(5.41)
|a−3/2 aτ | . (k∇ξk2L4 + kξkH 1 )kξkH 2 .
(5.42)
This together with (5.33) gives
36
This implies |a(τ ) − n| ≤ 41 , and
|a(τ )
− 12
− 12
− a(0)
1
|≤
2
Z
τ
Z
− 23
|a(s) aτ (s)|ds .
0
Z τ
1
≤ Λk (ξ0 )
e− 2 s 1.
τ
Λk (ξ)ds
0
(5.43)
0
Next, by (5.34),
Z
τ
Z
τ
λ(s)Λk (ξ)ds
|zτ (s)|ds .
0
0
Z τ
1
≤ Λk (ξ0 )
e−(n+ 2 )s 1.
|z(τ ) − z(0)| ≤
(5.44)
0
Rt
Rt
Observe that λ20 − λ(t)2 = 2 0 a(τ (s))ds. Let t∗ be the zero of the function λ20 − 2 0 a(τ (s))ds. Since
|a(τ ) − n| ≤ 21 , we have t∗ < ∞ and λ(t) → 0 as t → t∗ . Similarly to (5.43), we know that
Z τ2
1
−1/2
−1/2
|a(τ2 )
− a(τ1 )
|.
e− 2 s ds → 0,
τ1
1
as τ1 , τ2 → ∞. Hence there exists a∗ > 0, such that |a(τ ) − a∗ | . e−(1− 2n )τ . Similar arguments show
Rt
1
1
that there exists z∗ ∈ Rn+1 such that |z(τ ) − z∗ | . e−(n+ 2 − 2n )τ . Then λ2 = λ20 − 2 0 a(τ (s))ds =
R
R t∗
R t ds
t
ds
2 t a(τ (s))ds = 2a∗ (t∗ − t) + o(t∗ − t). The latter relation implies that τ = 0 λ(s)2 = 2a1∗ 0 (t∗ −s)(1+o(1))
p
1
1
1
1
1
1
−(1− 2n
)τ
and
= O((t∗ −t) 2a∗ (1− 2n ) ). So λ(t) = 2a∗ (t∗ − t)+O((t∗ −t) 2 + 2a∗ (1− 2n ) ), ρ(ω, τ (t)) =
q therefore e
1
n
2n .
The latter inequality with k = s, together with
a(τ (t)) + ξ(ω, τ (t)), and kξ(ω, τ (t))kH k . (t∗ − t)
estimates on a, z and λ obtained above, together with (5.7) and (5.9), proves Theorem 36. 6
Stability of elementary solutions II. Neckpinching
We consider boundariless hypersurfaces, {Mt | t ≥ 0}, in Rn+2 , given by immersions ψ(·, t) : Sn × R → Rn+1 ,
evolving by the mean curvature flow Then ψ satisfy the evolution equation:
∂ψ
= −H(ψ)ν(ψ),
∂t
(6.1)
where ν(x) and H(x) are the outward unit normal vector and mean curvature at x ∈ Mt , respectively.
We are interested in the situation where, as the result MCF evolution the surface develops and narrow
neck, which eventually pinches off - the neckpinching. This can be defined formally as follows. Assume the
solution, S(t), of the MCF is an immersion of the cylinder C n+1 . We say S(t) undergoes a neckpinching at
time t∗ and at a point x∗ if dist(p, x−axis) > 0 ∀p ∈ S(t), ∀t < t∗ and lim inf t→t∗ dist(ψ(x, ω), x−axis) >
0 ∀p 6= p∗ and = 0, for p = p∗ .
There are two distinct scenarios one can consider depending on whether the initial surface is compact
or not (the periodic case is equivalent to the compact, see more below). We interesting in the second,
substantially more difficult case. We consider initial surfaces close to cylinders. We encounter two new
elements as compared to the collapse to a point:
37
(i) cylinders have a larger group of transformations compared to spheres: besides of shifts there are
rotations of the axis (tilting),
(ii) the initial surface is non-compact.
The latter fact makes the problem much more difficult. In the rest of this section we present the latest
result on the neckpinch for a fixed cylinder axis, explain its proof and discuss possible extension to the case
when the axis is allowed to move (tilt).
6.1
Results
Write points in Rn+2 as (x, ω) = (x, ω 1 , . . . , ω n+1 ). (We label components of points of Rn+2 as (x0 , x1 , . . . , xn+1 ).)
2
The round cylinder R × Sn is naturally embedded as C n+1 = {(x, w) : |ω| = 1} ⊂ Rn+2 . We say a surface
n+1
given by immersion ψ is a (normal) graph over the cylinder C
, if there a function u : C n+1 → R+ (called
the graph function) s.t. ψ = ψu , where
ψu : (x, w) 7→ (x, u(x, ω)ω).
(6.2)
Initial conditions. We consider initial surfaces, S0 , given by immersions ψu0 , which are graphs over the
cylinder C n+1 , with the graph functions u0 , which are positive, have, modulo small perturbations, global
minima at the origin, are slowly varying near the origin and are even w.r. to the origin.
More precisely, we assume that (a) u0 (x, ω) satisfies, for (k, m, `) = (3, 0, ??), ( 11
10 , 0, ??), (1, 2, ??) and
(2, 1, ??), the estimates
2
k+m+1
2
1
0x
) 2 kk,m,` ≤ Cε0
ku0 − ( 2n+ε
2ς0
u0 (x, ω) ≥
√
,
(6.3)
n,
weighted bounds on derivatives to the order 5.
symmetries: u0 (x, ω) = u0 (−x, ω), either u0 (x, ω) = u0 (x), or n = 2 and u0 (x, θ) = u0 (x, θ + π).
The last condition fixes the the point of the neckpinch and the axis of the cylinder. Here we used the norms
kφkk,m,` := hxi−k ∂xm ∇` φL∞ .
Neckpinching profile.
Theorem 40 ([28, 29]). Under initial conditions described above and for ε0 sufficiently small, we have
(i) there exists a finite time t∗ such that inf u(·, t) > 0 for t < t∗ and limt→t∗ u(·, t) > 0, for x 6= 0 and
= 0, for x = 0;
(ii) there exist C 1 functions λ(t), c(t) and b(t) such that
s
2n + b(t)y 2
ku(x, ω, t) − λ(t)[
kk,m,` . b2 (t),
c(t)
(6.4)
38
for (k, m, `) =?? and with y := λ−1 (t)x. Moreover, the parameters λ(t), b(t) and c(t) satisfy the
estimates
1
λ(t) = s 2 (1 + o(1));
Here s := t∗ − t and λ0 = √
s := t∗ − t.
b(t)
=
− lnn|s| (1 + O( | ln1|s| |3/4 ));
c(t)
=
1+
1
ε
2ς0 + n0
1
ln |s| (1
(6.5)
+ O( ln1|s| )).
, with ς0 , ε0 > 0 depending on the initial datum and o(1) is in
(iii) if u0 ∂x2 u0 ≥ −1 then there exists a function u∗ (x, ω) > 0 such that u(x, ω, t) ≥ u∗ (x) for x 6= 0 and
t ≤ t∗ .
We outline the main steps in the proof of this result.
6.2
Rescaled surface
Let Mt denote a smooth family of smooth surfaces, given by immersions ψ(·, t) : R × Sn → Rn+1 and
evolving by the mean curvature flow, (6.1). It is convenient to rescale it as follows
ϕ(y, ω, τ ) = λ−1 (t)ψ(x, ω, t),
(6.6)
where y = y(x, t) and τ = τ (t), with
y(x, t) := λ−1 (t)[x − x0 (t)]
Z
and
τ (t) :=
t
λ−2 (s) ds,
(6.7)
0
respectively, where x0 (t) marks the center of the neck. If the point of the neckpinch is fixed by a symmetry
of the initial condition, then we set x0 (t) = 0. We derive the equation for ϕ. For most of the derivation, we
suppress time dependence.
−1
Let λ˙ = ∂λ
H(ϕ) and ν(ψ) = ν(ϕ) (see (4.3) - (4.5)),
∂t . Using ψ = λϕ and the relations H(ψ) = λ
∂ϕ ∂ϕ
∂y ∂ϕ
∂y
−2 ∂ϕ
−1
˙
˙
˙ −1 , we obtain from
= λϕ + λ ∂t , ∂t = λ ∂τ + ∂t ∂y and ∂t = −λλ y, and the definition a := −λλ
(6.1) the equation for ϕ, λ, z and g:
∂ψ
∂t
∂ϕ
= −H(ϕ)ν(ϕ) + aϕ − ay∂y ϕ.
∂τ
The equation (6.8) has the static solutions: a = a positive constant, ϕ(y, w) = (y,
6.3
(6.8)
pn
a w).
Linearized map
Consider
p n the equation (6.8). This equation has the static solutions ϕ(y, w) = ψρ (y, w) = (y, ρw), with
ρ =
a . They describe the cylinders of the radii ρ, shifted by z and rotated by g. We would lie to
39
investigate the linearized stability of these solutions. To this end, we linearize the r.h.s. of (6.8) around
these static solutions, considering only variations in the normal direction, to obtain (see (4.25))
La := −∂y2 + ay∂y +
a
(∆Sn + 2n)
n
where ∆Sn is the Laplace-Beltrami operator on Sn . We have already encountered this operator in Section 2, as
the normal hessian of modified volume functional around a cylinder. (Recall that the operator − na (∆Sn +2n),
as the normal hessian of modified volume functional around a sphere. It played an important role in our
study of spherical collapse.)
For a fixed a, properties of the operator La were described in Section 2: it is self-adjoint on the Hilbert
a 2
space L2 (R × Sn , e− 2 y dydw) and its spectrum consists of the eigenvalues [k − 2 + n1 l(l + n − 1)2 ]a, k =
0, 1, 2, 3, . . . ; ` = 0, 1, 2, . . . , of the multiplicities ml , with the normalized eigenvectors given by
φk,l,m,a (y, w) := φk,a (y)Ylm (w),
(6.9)
where, recall, φk,a (y) are the normalized eigenvectors of the linear operator −∂y2 − ay∂y corresponding
to the eigenvalues ka : k = 0, 1, 2, 3, . . . , and Ylm are the eigenfunctions of −∆Sn corresponding to the
eigenvalue l(l + n − 1) (the spherical harmonics). This implies that the only non-positive eigenvalues of La
are −2a, −a, 0, corresponding to the eigenvectors, labeled by (k, l) = (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), with
multiplicities 1, n + 2; n + 2 (so that the total multiplicity of the non-positive eigenvalues is 2n + 5). The
first three eigenfunctions of the operator −∂y2 − ay∂y , corresponding to the eigenvalues, −2a, −a, 0, are
φ0,a (y) = (
a 1
a 1√
a 1
) 4 , φ1,a (y) = ( ) 4 ay, φ2,a (y) = ( ) 4 (1 − ay 2 ),
2π
2π
2π
(6.10)
and first n + 2 eigenfunctions of the operator −∆Sn , corresponding to the first eigenvalue 0, and the second
eigenvalue n are 1 and w1 , · · · , wn+1 , respectively. Hence, La has
1
a 4
) ((k, l) = (0, 0)), due
• the eigenvalue −2a of the multiplicity 1 with the eigenfunction φ0,0,0,a (y) = ( 2π
to scaling of the transverse sphere;
1
a 4 m
• the eigenvalue −a of the multiplicity n + 1 with the eigenfunctions φ0,1,m,a (y) = ( 2π
) w , m =
1, . . . , n + 1 ((k, l) = (1, 1)), due to transverse translations;
a 14 √
) aywm , m =
• the eigenvalue 0 of the multiplicity n + 1 with the eigenfunctions φ1,1,m,a (y) = ( 2π
1, . . . , n + 1 ((k, l) = (0, 1)), due to rotation of the cylinder;
a 14 √
• the eigenvalue −a of the multiplicity 1 with the eigenfunction φ1,0,0,a (y) = ( 2π
) ay ((k, l) = (1, 0)),
due to translations of the point of the neckpinch on the axis of the cylinder;
1
a 4
• the eigenvalue 0 of the multiplicity 1 with the eigenfunction φ2,0,0,a (y) = ( 2π
) (1−ay 2 ) ((k, l) = (2, 0)),
due to shape instability.
The modes φ1,a and φ2,a are due to the location of the neckpinch (the shift in the x direction) and
its shape (see [28] and below). To see how they originate, we replace
p nthe cylindrical, static (i.e. y and
τ -independent) solution ϕ(y, w) = ψVa (y, w) = (y, ρa w), with Va =
a , to (6.8) with a constant, by a
modulated cylinder solution
r
2n + by 2
ϕab (y, w) = ψVab (y, w) = (y, Vab (y)w), with Vab (y) =
,
(6.11)
2a
40
which is derived later and which incorporates some essential features of the neckpinch. They are approximate
static solutions to (6.8) with a constant, −H(ϕab ) + aϕab · ν(ϕab ) ≈ 0. Differentiating it w.r.to a, x0 (see
(6.7)) and b and using that b is sufficiently small and therefore by 2 can be neglected in a bounded domain,
we arrive at vectors proportional to φ0,a , φ1,a and φ2,a .
Recall from Section 6.8 that the modes φk,l,m,a , (k, l, m) = (0, 1, 1), . . . , (0, 1, n + 1), are coming from the
transverse shifts, and φk,l,m,a , (k, l, m) = (1, 1, 1), . . . , (1, 1, n+1), (i.e. φ1,a (y)wm ' ywm , m = 1, . . . , , n+1),
from the rotations of the cylinder. Hence fixing the cylinder axis, eliminates these modes.
6.4
Equations for the graph function
Assume that for each t in the the existence interval, the transformed immersion ϕ(y, w, τ ), given in (6.6), is
a graph over the cylinder ϕ(y, w, τ ) = (y, v(y, w, τ )w). where y and τ are blowup variables defined by (6.7),
∂f
with x0 = 0. Then, similarly to the derivation of the equation (5.10) and using ∂f
∂t = ∂τ ∂t τ, for any
−2
−2 ˙
−2
˙ we see that
function f of the variables τ , ∂t τ = λ and ∂t y = −λ λxx = λ ayx, where, recall, a = −λλ,
(6.8) implies
∂τ v = G(v) − ay∂y v + av,
(6.12)
where the map G(v) given by
G(v) :=v −2 ∆v + ∂y2 v − nv −1 − f (v)
f (v) :=
2v −2 ∂y vh∇v, ∇∂y vi + (∂y v)2 ∂y2 v
v −4 h∇v, Hess v ∇vi + v −2 |∇v|2
+
,
1 + (∂y v)2 + v −2 |∇v|2
1 + (∂y v)2 + v −2 |∇v|2
(6.13)
with ∇ and Hess denoting the Levi-Civita covariant differentiation and hessian on a round Sn of radius one,
with the components ∇i and ∇i ∇j in some local coordinates, respectively.
Initial conditions for v. If λ0 = 1, then the initial conditions for u given in Theorem 40 implies that
there exists a constant δ such that the initial condition v0 (y) is even and satisfy for, (k, m, `) = (3, 0, ??),
( 11
10 , 0, ??), (1, 2, ??) and (2, 1, ??), the estimates
2
k+m+1
2
1
0y
kv0 (y, ω) − ( 2n+ε
) 2 kk,m,` ≤ Cε0
1− 1 ε
n 0
v0 (y, ω) ≥
√
,
n,
weighted bounds on derivatives to the order 5,
symmetries: v0 (y, ω) = v0 (−y, ω), v0 (y, θ) = v0 (y, θ + π).
6.5
Reparametrization of the solution space
(6.14)
41
Adiabatic solutions. The equation (6.12) has the following cylindrical, static (i.e. y and τ -independent)
solution
r
n
Va :=
and a is constant ⇐⇒ ucyl (t).
(6.15)
a
In the original variables t and x,p
this family of solutions corresponds to the cylinder (homogeneous) solution
X(t) = (x, R(t)ω), with R(t) = R02 − 2nt.
A larger family of approximate solutions is obtained by assuming that v is slowly varying and ignoring
∂τ v and ∂y2 v in the equation for v to obtain the equation
ayvy − av +
n
=0
v
(adiabatic or slowly varying approximation). This equation has the general solution
r
2n + by 2
,
(6.16)
Vab (y) :=
2a
p
pn
2
b
2π 14
√
with b ∈ R. Note also that for by 2 small, Vab (y) ≈ na (1 + by
4n ) =
a + 4a an ( a ) [φ0,a (y) + φ2,a (y)], i.e.
pn
the static solution a is modified by vectors along its unstable and central subspaces.
In what follows we take b ≥ 0 so that Vab is smooth. Note that Va0 = Va .
The rescaled equation (6.12) has two unknowns v and a. Hence we need an extra restriction specifying
a or λ.
Reparametrization of solutions. We want to parametrize a solution to (6.12) by a point on the manifold
Mas , formed by the family of almost solutions (6.16) of equation (6.12),
Mas := {Vab | a, b ∈ R+ , b ≤ },
and the fluctuation almost orthogonal to this manifold (large slow moving and small fast moving parts of
the solution):
v(w, y, τ ) = Va(τ )b(τ ) (y) + φ(w, y, τ ),
(6.17)
with
φ(·, τ ) ⊥ φka(τ ) in L2 (R × Sn , e−
a(τ ) 2
2 y
dydw), k = 0, 1, 2.
(6.18)
2
(Provided b is sufficiently small, the vectors φ0a = 1 and φ2a
√ = (1 − ay ) are almost tangent vectors to the
manifold, Mas . Note that φ is already orthogonal to φ1a = ay, since by our initial conditions the solutions
are even in y. For technical reasons, it is more convenient to require the fluctuation to be almost orthogonal
to the manifold Mas .)
As in the spherical collapse case, we consider (6.12) as an equation for a and v, or a, b and φ, and the
equation (6.18) leads to the additional equations on the parameters a, x0 (see (6.7)) and b.
As the space of fluctuations, we choose the space
X := {φ ∈ L∞ : kφkk,m,` < ∞, (k, m, `) = (3, 0, ??), (
11
, 0, ??), (1, 2, ??), (2, 1, ??)}.
10
6.6
42
Lyapunov-Schmidt splitting (effective equations)
In what follows we consider only surfaces of revolution. The decomposition (6.17) becomes
v(y, τ ) = Va(τ ),b(τ ) (y) + φ(y, τ ),
(6.19)
Substitute (6.19) into (MCF) to obtain
∂τ φ = −Lab φ + Fab + Nab (φ)
(6.20)
where Lab is the linear operator given by
Lab := −∂y2 + ay∂y − 2a +
aby 2
2+
by 2
n
and the functions F (a, b) and N (a, b, φ) are defined as
Fab :=
1 2n + by 2 1
1
b3 y 4
y2
],
(
) 2 [Γ1 + Γ2
+
−
2
2a
2n + by 2
n (2n + by 2 )2
with
∂τ a
2a
(6.21)
+ 2a − 1 + nb ,
Γ1
:=
Γ2
:= −∂τ b − b(2a − 1 + nb ) −
Nab (ξ)
2a
2
:= − nv 2n+by
2ξ −
b2
n,
(6.22)
(∂y v)2 ∂y2 v
1+(∂y v)2 .
Here we ordered the terms in Fab according to the leading power in y 2 .
Remember that
φ(·, τ ) ⊥ 1, a(τ )y 2 − 1 in L2 (R, e−
a(τ ) 2
2 y
dy).
(6.23)
2
Project the above equation on 1, a(τ )y − 1 =⇒ the equations for the parameters a, b.
Estimating φ.
Let U (τ, σ) be the propagator generated by −Lab . By Duhamel principle we rewrite the
differential equation for φ(y, τ ) as
Z τ
φ(τ ) = U (τ, 0)φ(0) +
U (τ, σ)(F + N )(σ)dσ.
(6.24)
0
The key problem in estimating φ(y, τ ) is to estimate the propagator U (τ, σ). We have to use (6.23).
Proposition 41. There exist constants c, δ > 0 such that if b(0) ≤ δ, then for any g ⊥ 1, a(τ )y 2 −
a(τ ) 2
1 in L2 (R, e− 2 y dy) and τ ≥ σ ≥ 0, we have
khzi−3 U(τ, σ)gk∞ . e−c(τ −σ) khzi−3 gk∞ .
43
Estimating the linear propagator.
Lab = L0 + V, where
Recall U (τ, σ) is the propagator generated by −Lab . We write
1
L0 := −∂y2 + ay∂y − 2a, V (y, τ ) ≥ 0 and |∂y V (y, τ )| . b 2 (τ ).
(6.25)
(L0 is the Ornstein-Uhlenbeck generator related to the harmonic oscillator Hamiltonian.)
Denote the integral kernel of U(τ, σ) by K(x, y). We have the representation (see Appendix D)
K(x, y) = K0 (x, y)heV i(x, y),
(6.26)
where K0 (x, y) is the integral kernel of the operator e−(τ −σ)L0 and
Z R
τ
V
he i(x, y) = e σ V (ω(s)+ω0 (s),s)ds dµ(ω).
(6.27)
Here dµ(ω) is a harmonic oscillator (Ornstein-Uhlenbeck) probability measure on the continuous paths
ω : [σ, τ ] → R with the boundary condition ω(σ) = ω(τ ) = 0 and
(−∂s2 + a2 )ω0 = 0 with ω(σ) = y and ω(τ ) = x.
(6.28)
By a standard formula we have
1
K0 (x, y) = 4π(1 − e−2ar )− 2
√
ae2ar e
−a
(x−e−ar y)2
2(1−e−2ar )
,
where r := τ − σ. To estimate U (x, y) we use that, by the explicit formula for K0 (x, y) given above,
|∂yk K0 (x, y)| .
e−akr
(|x| + |y| + 1)k K0 (x, y),
(1 − e−2ar )k
and by an elementary estimate
1
|∂y heV i(x, y)| ≤ b 2 r.
(6.29)
Estimate for ear ≤ β −1/32 (τ ) (r := τ − σ) and then iterate using the semi-group property.
Now, (6.26) implies Equation ( 6.29) by the following lemma.
Lemma 42. Assume in addition that the function V (y, τ ) satisfies the estimates
1
V ≤ 0 and |∂y V (y, τ )| . β − 2 (τ )
(6.30)
where β(τ ) is a positive function. Then
Z R
τ
1
|∂y e σ V (ω0 (s)+ω(s),s)ds dµ(ω)| . |τ − σ| sup β 2 (τ )
σ≤s≤τ
Proof. By Fubini’s theorem
Z R
Z
Z
τ
V (ω0 (s)+ω(s),s)ds
σ
∂y e
dµ(ω) = ∂y [
0
τ
V (ω0 (s) + ω(s), s)ds]e
Rτ
σ
dµ(ω)
44
Equation ( 6.30) implies
Z τ
Rτ
1
|∂y
V (ω0 (s) + ω(s), s)ds| ≤ |τ − σ| sup β 2 (τ )|, and e σ V (ω0 (s)+ω(s),s)ds ≤ 1.
σ≤s≤τ
σ
Thus
Z
|∂y
e
Rτ
σ
1
2
dµ(ω)| . |τ − σ| sup β (τ )|
Z
1
dµ(ω) = |τ − σ| sup β 2 (τ )|
σ≤s≤τ
σ≤s≤τ
to complete the proof.
Derivation of Proposition 41 from (6.26) and (6.29). Bootstrap.
Let (a, b, φ) be the neck parametrization of the rescaled solution v(y, τ ). To control the function φ(y, τ ),
we use the estimating functions
Mk,m (T ) := max b−
k+m+1
2
τ ≤T
with (k, m) = (3, 0), ( 11
10 , 0), (2, 1), (1, 2). Let |M | :=
(τ )khyi−k ∂ym φ(·, τ )k∞ ,
P
i,j
M := (Mi,j ), (i, j) = (3, 0), (
(6.31)
Mi,j and
11
, 0), (1, 2), (2, 1).
10
(6.32)
Using Proposition 41 and a priori estimates obtained with help of a maximum principle (see Appendix
??), we find
Proposition 43. Assume that for τ ∈ [0, T ] and
1
|M (τ )| ≤ b− 4 (τ ), v(y, τ ) ≥
1√
2n, and ∂ym v(·, τ ) ∈ L∞ , m = 0, 1, 2.
4
Then there exists a nondecreasing polynomial P (M ) s.t. on the same time interval,
1
Mk,m (τ ) ≤ Mk,m (0) + b 2 (0)P (M (τ )),
(6.33)
Corollary 44. Assume |M (0)| 1. On any interval [0, T ],
1
|M (τ )| ≤ b− 4 (τ ) =⇒ |M (τ )| . 1.
The analysis presented above goes through also in the non-radially symmetric case, with the cylinder
axis fixed, but with one caveat. The feeder bounds have to be extended to the non-radially symmetric case
and new bounds on θ- and mixed derivatives, ∂yn ∂θm v(y, θ, τ ), have to be obtained. This requires additional
tools (differential inequalities for Lyapunov-type functionals.
6.7
General case. Transformed immersions
Let Mt denote a smooth family of smooth hypersurfaces, given by immersions ψ(·, t) and evolving by the
mean curvature flow, (6.1). Instead of the surface Mt , it is convenient to consider the new, rescaled surface
45
˜ τ = λ−1 (t)g(t)−1 (Mt − z(t)), where λ(t) and z(t) are some differentiable functions and g(t) ∈ SO(n + 2),
M
Rt
to be determined later, and τ = τ (t) := 0 λ−2 (s)ds. If we look for ψ as a graph over a cylinder, then we
have to rescale also the variable x along the cylinder axis, y = λ−1 (t)(x − x0 (t)) (see (6.7)). The new surface
is described by ϕ, which is an immersion of the fixed cylinder C n+1 , i.e. ϕ(·, τ ) : C n+1 → Rn+2 , given by
ϕ(y, w, τ ) = λ−1 (t)g −1 (t)(ψ(x, w, t) − z(t)).
where y = y(x, t) and τ = τ (t) are given by y = λ−1 (t)(x − x0 (t)) and τ =
(6.34)
Rt
0
λ−2 (s)ds.
We derive the equation for ϕ. For most of the derivation, we suppress the time dependence. Let λ˙ = ∂λ
∂t .
Using that X = z + λgϕ and the relations (4.3) - (4.5) , which imply H(ψ) = λ−1 H(ϕ) and ν(ψ) = gν(ϕ),
and using
∂ψ
˙
= z˙ + λgϕ
+ λgϕ
˙ + λg ϕ,
˙
∂t
we obtain from (6.1) the equation for Y , λ, z and g:
∂ϕ
= −H(ϕ)ν(ϕ) + (a − y∂y )ϕ − g −1 gϕ
˙ − λ−1 g −1 z.
˙
∂τ
(6.35)
The equation (6.35)
has the static solutions (a = a positive constant, z =constant, g =constant, x0 =constant,
p
Y (x, w) = (x, na w)).
Note that we do not fix λ(t) and x0 (t), z(t) and g(t) but instead consider them as free parameters to be
determined from the evolution of v in equation (6.35).
Remark 5. Another way to write (6.34) is ϕ = (Tλ Tz Tg ψ) ◦ Sλ , where Tλ , Tz and Tg are the scaling,
translation and rotation maps defined by Tλ ψ := λ−1 ψ, Tz ψ := ψ − z, z ∈ Rn+2 , and Tg ψ := g −1 ψ,
Rt
g ∈ SO(n + 2), and acting pointwise and Sλ−1 : (x, ω, t) → (y = λ−1 (t)x, w, τ = 0 λ−2 (s)ds).
6.8
Translational and rotational zero modes
(do not probably need) Recall that the maps ψ = ψρzg , with z, g and ρ constant, describe the cylinders
of the radii ρ, shifted by z and rotated by g. They are static solutions of (6.35) with a constants, i.e. solve
−H(ψzgρ ) + aψzgρ · ν(ψρzg ) = 0,
(6.36)
p
provided ρ = na . Differentiating this equation w.r. to a, z i and g ib , b = n + 2, we arrive at the scaling,
translational and rotational modes. We expect that these are exactly the modes mentioned in Section 2 and
Subsection 6.3.
6.9
Collapse center and axis
In this section, extending [56], we introduce a notion of the ’centre’ and ’axis’ of a surface, close to the round
cylinder C n+1 , and show that such centre and axis exist. We define the cylinder axis, A, by a unit vector
α ∈ Sn+1 so that A := Rα. Moreover, we identify the axis vector α with an element g ∈ SO(n+2)/SO(n+1)
s.t. gen+2 = α.
46
Our eventual goal will be to show that the centers z(t) and axis vectors α(t) of the solutions Mt to the
MCF converge to the collapse point, z∗ , and axis vector, α∗ . For a boundaryless surface S, given by an
immersion ψ : C n+1 → Rn+2 , we define the center, z, and the axis vector, α (or g s.t. gen+2 = α), by the
relations
Z
(P ⊥ g −1 (ψ(x, w) − z) · w)xk wi = 0, i = 1, . . . , n + 1, k = 0, 1,
(6.37)
C n+1
where P ⊥ is the orthogonal projection to the subspace {(0, w)} ⊂ Rn+2 . Here and in what follows, the
integrals over the set C n+1 (or R × Sn ), which do not specify the measure, are taken w.r.to the measure
a 2
a 2
e− 2 x dwdx or e− 2 y dwdy. These are 2n + 2 equations for the 2n + 2 unknowns z ∈ Ran Pg⊥ and g ∈
SO(n+2)/SO(n+1), where SO(n+1) is the group of rotations of the subspace {(w, 0)} ⊂ Rn+2 . Note that the
rotated, shifted and dilated (transformed) cylinders, Cλzg , defined by the immersions Xλzg : C n+1 → Rn+2 ,
given by
ψλzg (x, w) := z + g(x, λw),
(6.38)
satisfy (6.37), which justifies our interpretation of its solutions as the centre and axis vector of a surface.
a
2
Denote by H 1 (C n+1 , Rn+2 ) the Sobolev space with the measure e− 2 x dwdx. We introduce the following
Definition 5. A surface M, given by an immersion ψ : C n+1 → Rn+2 , is said to be H 1 −close to a
transformed cylinder Cλzg , iff ϕ := λ−1 g −1 (X − z) ∈ H 1 (C n+1 , Rn+2 ) close, in the H 1 (C n+1 , Rn+2 )-norm,
to the identity 1 : C n+1 → C n+1 .
Proposition 45. Assume a surface M, given by an immersion ψ : C n+1 → Rn+2 , is H 1 −close to a
+
¯
¯ ∈ Rn+1 and g¯ ∈ SO(n + 2). Then there exists g ∈ SO(n +
transformed cylinder Cλ¯
¯ zg
¯ , for some λ ∈ R , z
⊥
2)/SO(n + 1) and z ∈ Ran Pg such that (6.37) holds.
Proof. By replacing ψ by ψ new , if necessary, we may assume that z¯ = 0, g¯ = 1 and λ = 1. The relations
(6.37) are equivalent to the equation F (ψ, z, g) = 0, where
F (ψ, z, g) = (Fik (ψ, z, g), i = 1, . . . , n + 1, k = 0, 1),
with Fik (ψ, z, g) equal to the l.h.s. of (6.37). Clearly F is a C 1 map from H 1 (C n+1 , Rn+2 ) × Rn+1 × SO(n +
2)/SO(n + 1) to R2n+2 . We notice that F (1, 0, 1) = 0. We solve the equation F (ψ, z, g) = 0 near (1, 0, 1),
using the implicit function theorem. To this end we have to calculate the derivatives ofRF w.r.to z and g at
ψ = 1, z = 0, g = 1. The derivatives with respect to z i are easy ∂zi Fjk |z=0,g=1 = − C n+1 wi wj xk , which
gives
∂zi Fjk |z=0,g=1 = −
1
|Sn |δij δk,0 ,
n+1
(6.39)
for i, j = 1, . . . , n + 1, k = 0, 1. To calculate the derivatives of F w.r.to g, we write g ∈ SO(n + 2)/SO(n + 1)
as a product of rotations g ab in two-dimensional planes, i.e. involving only variables xa and xb , with a = 0
and b = 1, . . . , n + 1, and denote by ∂gab the derivative w.r.to the angle of the corresponding rotation.Using
that ∂gab g −1 |z=0,g=1 = −àb , the generator of the rotation in the ab−plane and using that 1(x, w) = (x, w),
we calculate
Z
∂gab Fjk |ψ=1,z=0,g=1 = −
(àb (w, x) · (0, w))ω j xk .
(6.40)
C n+1
47
P
We see that, if a, b ∈ {1, . . . , n + 1}, then àb (x, w) · (0, w) = −wb wa + wa wb + i6=a,b (wi )2 and therefore
R
(` (x, w) · (0, w))ω j = R0. Furthermore, if a = 0 and b = 1, . . . , n + 1, then àb (x, w) · (0, w) = xwb +
Sn ab
P
i 2
j
i6=a,b (w ) and therefore Sn (àb (x, w) · (0, w))ω = δb,j x. This gives, for a = 0,
Z
∂gai Fjk |ψ=1,z=0,g=1 = cn
a
2
x2 e− 2 x dxδij δk,1 ,
(6.41)
R
for i, j = 1, . . . , n + 1, k = 0, 1. Hence dF is invertible and we can apply implicit function theorem to show
that for any ψ close to 1, there exists z and g, close to 0 and 1, respectively, such that F (ψ, z, g) = 0.
˜ τ , i.e. the immersions (6.34), are graphs over the round
Assume that the transformed surfaces, M
n+1
cylinder C
determined by the functions v : C n+1 × [0, T ) → R+ , z(t) ∈ Rn+2 , g(t) ∈ SO(n + 2), as
ϕ(y, ω, τ ) = (y, v(y, w, τ )w), or
ψ(x, w, t) = z(t) + λ(t)g(t)(y, v(y, w, τ )w).
(6.42)
¯
Assume also there are functions z¯(t) ∈ Rn+2 , g¯(t) ∈ SO(n + 2)/SO(n + 1) and λ(t)
∈ R+ , s.t. ψ(·, t) is
1
H −close to a transformed cylinder Cλ¯
¯ zg
¯ in the sense of the definition (5). Then Proposition 45 implies that
there exists z(t) ∈ Rn+2 and g(t) ∈ SO(n + 2)/SO(n + 1), s.t. (6.37) holds. If ψ is a (λ, z, g)−graph over
C n+1 , i.e. it is of the form (6.42), then v satisfy
Z
v(y, w, τ )y k wj = 0, j = 1, . . . , n + 2, k = 0, 1.
(6.43)
C n+1
To apply Proposition 45 to the immersion ψ(·, t) : C n+1 → Rn+2 , solving (6.1), we pick z¯(t) to be a
piecewise constant function constructed iteratively, starting with z¯(t) = 0 and α
¯ (t) = e1 for 0 ≤ t ≤ δ
for δ sufficiently small (this works due to our assumption on the initial conditions), and z¯(t) = z(δ) and
α
¯ (t) = α
¯ (δ) for δ ≤ t ≤ δ + δ 0 and so forth. This gives z(t) ∈ Rn+1 and g(t) ∈ O(n + 2), s.t. (6.43) holds.
6.10
Equations for the graph function
Assume that for each t in the the existence interval, the transformed immersion ϕ(y, w, τ ), given in (6.34),
is a graph over the cylinder
ϕ(y, w, τ ) := λ−1 (t)g(t)−1 (ψ(x, w, t) − z(t)) = (y, v(y, w, τ )w),
(6.44)
where y and τ are blowup variables defined by (6.7). Then, similarly to the derivation of the equation
∂f
−2
(5.10) - (5.11) or (6.12), and using ∂f
and
∂t = ∂τ ∂t τ, for any function f of the variables τ , ∂t τ = λ
˙ − x0 ) − λ−1 x˙ 0 = λ−2 (ay − λ−1
∂t y = −λ−2 λ(x
∂x0
),
∂τ
˙ we see that (6.35) implies (cf. (6.12))
where, recall, a = −λλ,
∂τ v = G(v) − a(y∂y − 1)v − λ−1
∂x0
∂g
∂z
∂y v − g −1 v − λ−1 g −1
∂τ
∂τ
∂τ
(6.45)
48
Rt
∂z
where ∂τ
be the τ -derivative of z(t(τ )), etc, with t(τ ) the inverse function of τ (t) = 0 λ−2 (s)ds, and the
map G(v) given by (5.11). For v have the orthogonal decomposition (6.17) - (6.18). This together with
a(τ ) 2
(6.43) implies that φ(·, τ ) satisfy (again in L2 (R × Sn , e− 2 y dydw))
φ(·, τ ) ⊥ 1, y, y 2 , y k wj , j = 1, . . . , n + 1, k = 0, 1.
(6.46)
Remark 6. Other approaches: 1) Write a point, p, on the hypersurface Mt as say p = q + xω + u(x, ν)ν,
where q gives the shift of the cylinder axis, ω is the unit vector in the direction of the cylinder axis, ω · q = 0,
ν is the outward unit normal to the cylinder surface. To find the equation for u, we compute p˙ · νMt =
P4
˙ for some γi and for ϕ1 = cos θ, ϕ1 = cos θ, ϕ2 = sin θ, ϕ3 = x cos θ, ϕ4 = x sin θ, while
i=1 γi ϕi + γ0 u,
v(y, θ, τ ) is now decomposed as v(y, θ, τ ) = Va(τ ),b(τ ) + φ(y, θ, τ ), with φ ⊥ {1, y, y 2 }. One can also consider
a moving frame (e1 , e2 , e3 ), with e3 = ω, so that ν = e1 cos θ + e2 sin θ.
2) Write v(y, θ, τ ) = Vã(τ ),b(τ ) +φ(y, θ, τ ), where Vã(τ ),b(τ ) = Va(τ ),b(τ ) +β0 (τ )y+β1 (τ ) cos θ+β2 (τ ) sin θ+
β3 (τ )y cos θ + β4 (τ )y sin θ and φ ⊥ {1, y, y 2 , cos θ, sin θ, y cos θ, y sin θ}. In other words, one ”dresses” up the
main, finite dimensional term absorbing into it all the neutral-unstable modes (singular, secular behaviour).
7
Volume preserving mean curvature flow
The volume preserving mean curvature flow (VPF) is a family {St ; t ≥ 0} of smooth closed hypersurfaces
in Rn+1 , given say by immersions ψ, satisfying the following evolution equation:
¯ − H,
∂t ψ N = H
N
(7.1)
N
where ∂t ψ denotes the normal velocity of St at time t, ∂t ψ = h∂t ψ, νi, where ν is the unit normal vector
¯ = H(t)
¯
field on St , H = H(t) stands for the mean curvature of St and H
is the average of the mean curvature
on St , i.e.,
R
Hdσ
¯
, t ≥ 0.
(7.2)
H := RSt
dσ
St
As an initial condition, we consider a simple hypersurface S0 in Rn+1 , with no boundary in Ω (e.g. either
entirely Ω or ∂S0 ⊂ ∂Ω) given by an immersion x0 .
Like the MCF, the VPF is invariant under rigid motions (translations and rotations) and appropriate
scaling. Moreover, it shrinks the area, A(ψ) = V (ψ), of the surfaces, but, unlike the MCF, the VPF preserves
the enclosed volume, Vencl (ψ). As the result, it has stationary solutions - the Euclidean spheres (for closed
surfaces) and cylinders for surfaces with flat boundaries. We summarize these properties as properties:
• (8.3) is invariant under rigid motions of the surface, i.e. ψ 7→ Rψ + a, where R ∈ O(n + 1), a ∈ Rn+1
and ψ = ψ(u, t) is a parametrization of St , is a symmetry of (8.3).
• (8.3) is invariant under the scaling x 7→ λx and t 7→ λ−2 t for any λ > 0.
• (8.3) is volume preserving.
• (8.3) is area shrinking. Actually,
Z
Z
Z
d
¯ − H 2 )dσ = −
¯ − H)2 dσ ≤ 0.
V (ψ(t)) =
Hh∂t ψ, νidσ =
(HH
(H
dt
St
St
St
49
• Static solutions of (8.3) are surfaces of constant mean curvature (CMC), H = h.
• (8.3) is a gradient flow for the area functional on closed surfaces with given enclosed volume.
The first two statements are proven as for the MCF. The third one follows from
R
¯ − H)dσ = 0. The fifth one is obvious.
(H
St
d
dt Vencl (ψ(t))
=
R
St
h∂t ψ, νidσ =
To prove
R the last Rstatement √we use the formula (1.9), which we rewrite in the present notation as
dV (ψ)ξ = ψ Hν · ξ = U Hν · ξ gdn u, and the fact that ξ is a vector field for deformations with a fixed
enclosed volume, i.e. it is a tangent vector field to the manifold Xc := {ψ : U → Rn+1 , Vencl (ψ) = c} (we
do not specify the topology
here, we could take the topology of differentiable or Sobolev functions), and
R
therefore it satisfies ψ ξ · ν = 0. Indeed, let ψs be a family of constant enclosed volume surfaces deforming
ψ and let ξ be the corresponding vector field at s = 0, i.e. ξ = ∂s ψs s=0 . Since Vencl (ψs ) = c, we have
dVencl (ψ)ξ = ∂s Vencl (ψs )s=0 = 0. Hence the result follows from the formula
Z
dVencl (ψ)ξ =
ξ · ν.
ψ
This formula can be proven by either considering an infinitesimal change Rin the enclosed Rvolume under the
1
1
div x = n+1
x · ν, where Ω is
variation of S or by using that, by the divergence theorem, Vencl (ψ) = n+1
Ω
S
domain enclosed by the surface S, described by the immersion ψ,
and
then
differentiating
the
latter integral,
R
see Appendix A.3 (to be added). Conversely, if f satisfies S f = 0, then there is a volume preserving
normal variation with the vector field f ν. This implies that
Z
Tθ Xc = {ξ : S → Rn+1 , ξ · ν = 0}.
S
R
1
Now, for an arbitrary normal vector field η = f ν, the vector field ξ := (f − f¯)ν, where f¯ := |S|
f,
S
R
R
R
R
¯
satisfies S ξ · ν = 0 and therefore we have dV (ψ)ξ = ψ Hν · ξ = ψ H(f − f ) = ψ (H − H)f . This shows
that the L2 −gradient of V (ψ) on closed surfaces with given enclosed volume is H − H.
The third and forth properties suggests that, as t → ∞, St converges to a closed surfaces with the
smallest surface area for a given enclosed volume. The limitting surface must be a static solution to the
VPF and therefore, by the fifth property, it is a surface of constant mean curvature (CMC). This leads to
the isoperimetric problem:
• minimize the area V (ψ) given the enclosed volume Vencl (ψ).
Namely, we have
Proposition 46. (i) Minimizers of the area A(ψ) for a given enclosed volume Vencl (ψ) are critical points
of the area functional A(ψ) on space of immersions with the given enclosed volume Vencl (ψ).
(ii) The (Euler-Lagrange) equation for these critical points is exactly the CMC equation H = h.
(iii) These critical points are critical points of the functional Vh (ψ) := V (ψ) − hVencl (ψ), where h is determined by c = Vencl (ψ) and vice versa.
50
Proof. The first and third statements are standard results (the third statement is follows
Lagrange
R from the
√ n
gd
u and
multiplier theorem).
The
second
statement
follows
from
the
relations
dV
(ψ)ξ
=
Hν
·
ξ
U
R
dVencl (ψ)ξ = St hξ, νidσ and the definition of Vh (ψ).
Thus finding stationary solutions to the VPF is the same as finding closed CMC surfaces. This leads to
the following problems: (a) Find CMC surfaces and (b) determine their stability w.r. to the VPF.
However, we expect that the VPF converges to a minimal CMC surface (i.e. one solving the isoperimetric
problem), not just to a CMC surface. For such we have an additional characterization, which is a standard
fact from the variational calculus (see e.g. [36]):
Proposition 47. If ϕ minimizes the area V (ψ) for a given enclosed volume Vencl (ψ), then HessN V (θ) ≥ 0
on Tθ Xc .
The property of minimal CMC surfaces isolated in this proposition plays an important role in their
analysis and deserves the name. We say that a CMC surface θ is stable iff HessN V (θ) ≥ 0 on Tθ Xc . (In
dynamical systems and partial differential equations, this notion is called (weak) linear or energetic stability.)
The next result illustrate an importance of the notion of stability and is one of the first of the slue of
similar results.
Theorem 48. A compact, stable CMC surface is a sphere.
Before proceeding to the proof of this theorem, we review some spectral theory of the normal hessian,
HessN V (ϕ), weich, recall, is defined as the hessian w.r.to normal variations, η = f ν, i.e HessN V (ϕ)f =
Hess V (ϕ)(f ν).
Theorem 49. The normal hessian, HessN V (ϕ), of the modified volume functional V (ϕ), at a CMC ϕ, has
the eigenvalue 0 of
• the multiplicity n + 1 + 12 n(n − 1) (+1, if h = 0), with the eigenfunctions ν j (ϕ), j = 1, . . . , n + 1, and
hσj ϕ, ν(ϕ)i, j = 1, . . . , 12 n(n − 1), where σj are generators of the Lie algebra of SO(n + 1) (Jacobi
fields), if ϕ is not spherical or axi-symmetric;
• the multiplicity 2n, if ϕ = cylinder,
• the multiplicity 0 (i.e. 0 is not an eigenvalue), if ϕ = sphere.
Moreover, if h 6= 0 and
R
S
hϕ, ν(ϕ)i > 0, then the operator dN H(ϕ) has a negative spectrum.
Proof. As above, we use the important property that the CMC equation H(ϕ) = h breaks the scaling,
translational and rotational symmetry, (4.3) - (4.5), which we recall here:
H(λϕ) = λ−1 H(ϕ), ∀λ ∈ R+ ,
H(ϕ + h) = H(ϕ), ∀h ∈ R
n+1
(7.3)
,
(7.4)
H(gϕ) = H(ϕ), ∀g ∈ O(n + 1).
(7.5)
51
Now, if an immersion ϕ satisfies the CMC equation H(ϕ) = h, then by (7.4), that H(ϕ + sz) = h and
any constant vector field z. Differentiating this equation w.r.to s at s = 0, we obtain dH(ϕ)z = 0. Now,
choosing ξ to be equal to the tangential projection, z T , of z, and subtracting the equation (4.1) from the
last equation, we find dH(ϕ)hz, ν(ϕ)iν(ϕ) = 0, which together with (4.2) gives the second statement.
Finally, we differentiate the equation H(g(s)ϕ) = h, where g(s) is a one-parameter subgroup of O(n + 1),
w.r.to s at s = 0, h to obtain dH(ϕ)σϕ = 0, where σ denotes the generator of g(s). Now, choosing ξ in
(4.1) to be equal to the tangential projection, (σϕ)T , of σϕ, and subtracting the equation (4.1) from the last
equation, we find dH(ϕ)hσϕ, ν(ϕ)iν(ϕ) = 0, h which together with (4.2) gives the third statement. Finally,
an explicit and elementary computation performed below shows that the normal hessian, HessN
sph V (ϕ) on a
sphere has no rotational eigenfunctions.
For the second statement, we show that, if an immersion ϕ satisfies the CMC equation H(ϕ) = h, then
dN H(ϕ)hϕ, ν(ϕ)i = −h.
(7.6)
Indeed, by the CMC equation H(ϕ) = h and (7.3), we have H(λϕ) = λ−1 h for any λ > 0. Differentiating
this equation w.r.to λ at λ = 1, we obtain dH(ϕ)ϕ = −h. Next, choosing ξ to be equal to the tangential
projection, ϕT , of ϕ, and subtracting the equation (4.1) from the last equation, we find dH(ϕ)hϕ, ν(ϕ)iν(ϕ) =
−h. Since by the definition (4.2), dH(ϕ)f ν(ϕ) = dN H(ϕ)f and Hess V (ϕ) = dH(ϕ), which proves the desired
statement.
R
The relation (7.6) implies that, if S hϕ, ν(ϕ)i > 0, then hdN H(ϕ)g, gi < 0, where g := hϕ, ν(ϕ)i, and
therefore the operator dN H(ϕ) has a negative spectrum.
The explicit expression for the normal hessian, HessN V (ϕ) was computed in Theorem 30. We recall it
here
HessN V (ϕ) = −∆ − |W |2 .
(7.7)
Proof
48. Let g := hϕ, ν(ϕ)i (the support function of ϕ). We write g = g ⊥ +¯
g , where, recall, f¯ :=
R of Theorem
1
⊥
N
N
|S| S f and g is defined by this relation, and use the self-adjointness of d H(ϕ), to compute hd H(ϕ)g, gi =
N
⊥ ⊥
N
N
N
hd H(ϕ)g , g i + 2hd H(ϕ)¯
g , gi − hd H(ϕ)¯
g , g¯i. This together with (7.6), d H(ϕ)¯
g = −|W |2 g¯ (by (7.7))
and the stability of ϕ gives
−h¯
g = hdN H(ϕ)g ⊥ , g ⊥ i + 2hdN H(ϕ)¯
g , gi − hdN H(ϕ)¯
g , g¯i
≥ −2h|W |2 g¯, gi + h|W |2 g¯, g¯i.
Now, we have h|W |2 g¯, gi = g¯h|W |2 , gi = g¯g|W |2 , where A := hA1, 1i, and similarly, h|W |2 g¯, g¯i = g¯2 |W |2 .
The last three relations give −h¯
g ≥ −2¯
g g|W |2 + g¯2 |W |2 , which implies
−h ≥ −2g|W |2 + g¯|W |2 .
Now, integrating (7.6) and using (7.7), we obtain g|W |2 = h, which together with the previous relation yields
h ≥ g¯|W |2 .
Next, by the Minkowski integral formula, gH = n, we have h¯
g = n, which together with the previous relation
yields
h2 ≥ n|W |2 .
52
On the other hand we have H 2 ≤ n|W |2 , with the equality valid iff S is a sphere. This together with
the previous relation implies h2 ≥ n|W |2 ≥ h2 and therefore n|W |2 = h2 , which shows that all principal
curvatures of ϕ are equal and therefore ϕ is a sphere.
Now, we define the notion of dynamic stability. We say that a CMC surface θ (which is a static solution
to (8.3)) is asymptotically stable iff there is ε s.t. for any initial condition ψ0 ∈ Xc , with c = Vencl (θ), ε−close
to θ in the H s norm, the solution ψ to the VPF (8.3), appropriately reparametrized converges, as t → ∞,
to the manifold Mc := {φ ∈ Xc : H(φ) = h(c)} (i.e. there is a family α = α(·, t) of reparametrizations, s.t.
dist(ψ ◦ α, Mc ) → 0).
There are also a weaker notion of the Lyapunov or orbital stability and of linearized (or energetic)
stability. The latter plays an important role in analysis of CMC surfaces and in proving asymptotic and
orbital stability. First, we give a weak form of this notion, which called the stability in the theory of minimal
and CMC surfaces, and later formulate a stronger version.
We also define a strong version of the notion of linear stability. We say that a CMC surface θ is
(strongly) linearly stable iff HessN V (θ) > 0 on Tθ Xcs Y , where YR := span{(ν j (ϕ))⊥ , j = 1, . . . , n + 1, and
1
f ) and, recall, σj are generators of the
hσj ϕ, ν(ϕ)i⊥ , j = 1, . . . , 21 n(n − 1)}. Here, f ⊥ := f − f¯ (f¯ := |S|
S
Lie algebra of SO(n + 1).
Asymptotic stability of spheres under the volume preserving flow was shown by J. Escher and G. Simonett
([25]) and D. Antonopoulou, G. Karali, I.M. Sigal ([3]).
∗∗∗∗∗
Remark: Another form of the VPF. Let ψ solve the rescaled MCF, (1.1). Rescale the surface ψ and
Rt
time t as (2.7), i.e. ϕ(u, τ ) := λ−1 (t)ψ(u, t), τ = 0 λ−2 (s)ds. Then the rescaled surface ϕ satisfies the
rescaled MCF, (2.6), which we reproduce here
(∂τ ϕ)N = −H(ϕ) + aϕN ,
(7.8)
˙
where (∂τ ϕ)N := h∂τ ϕ, ν(ϕ)i, ϕN := hϕ, ν(ϕ)i and a = −λλ.
Now we pick λ so that the enclosed volume
is preserved, i.e. Vencl (ϕ) = constant. RNow, Vencl (ϕ) = Vencl (λ−1 ψ) = λ−n Vencl
R (ψ) and therefore 0 =
˙ −1 Vencl (St ) −
∂t (λ−n V ol(ψ)) = λ−n (−nλ−1 Vencl (St ) + St h∂t ψ, νi) = λ−n (−nλλ
H(ψ), which gives a =
St
R
¯
H/nVencl (St ) =: H/n. This is related but not identical to (8.3).
St
8
Volume preserving mean curvature flow for submanifolds:
Trapped surfaces
We consider the volume preserving mean curvature flow (VPF) of closed hypersurfaces, S = {St ; t ≥ 0}
in the simplest non-euclidean space - the conformally euclidean one, (Rn+1 , e−2ϕ(x) ds2eucl ). Moreover, we
consider a slowly conformal factor ϕ(x) = ϕε (x), depending on some small parameter ε > 0 and satisfying
the estimates
ϕε (x)|ε=0 = 0,
and |∂xα ϕε (x)| . ε|α| .
(8.1)
53
One can consider general, slowly varying ambient metric, but the problems we are interested in are interesting
and rather non-trivial already for the conformal metric. We label all quantities pertaining to the metric
e−2ϕε (x) ds2eucl
(8.2)
by the subindex ε.
R
Let |S| := S 1 denote the surface area of a surface S. We denote by hf iS the average, hf iS :=
−2ϕε (x)
of f over S, and by hξ(x), η(x)iϕ = ξ(x) · η(x)e
the eucledian dot product.
, the conformal inner product in R
n+1
1
|S|
R
S
f,
, where ξ · η is
The VPF is a family St of surfaces, given by immersions ψ(·, t), satisfying, for t ≥ 0, the differential
equation,
∂t ψ N = hHε iSt − Hε ,
(8.3)
where ∂t ψ N is the normal velocity, ∂t ψ N := h∂t ψ, ν iϕ , where νε is unit normal on St , and Hε is the mean
curvature of ψ. (We suppress the time - dependence of the objects above.) Our goal is to investigate static
solutions of (8.3), i.e. constant mean curvature surfaces in (Rn+1 , e−2ϕε (x) dx2 ). Specifically, we would like
to investigate
1. Existence of static solutions,
2. Their stability/instability.
Note that a closed surface S, given by an immersion θ and satisfying the static VPF equation,
Hε (θ) = hHε iS ,
(8.4)
is a constant minimal curvature surface, satisfying the equation
Hε (θ) = h
for some constant h and vice versa. Let Sλ,z denote the sphere of radius λ, centred at z ∈ R
like to show that
(8.5)
n+1
. We would
1. There exists constant mean curvature surface close to a sphere Sλ,z iff z is a critical point of the scalar
curvature Rε of (Rn+1 , e−2ϕε (x) dx2 ).
2. The above constant mean curvature surface is stable iff z is a minimum of Rε .
More precisely,
Theorem 50. Assume (8.2) and (8.1). Then for each non-degenerate critical point, x∗ , of the scalar
curvature Rε of (Rn+1 , e−2ϕε (x) dx2 ) and for each λ > 0, there exist λ(ε) = λ + O(ε) ∈ R+ and x(ε) =
x∗ + O(ε) ∈ Rn+1 and a family of constant mean curvature surfaces (static solutions),
S (ε) = Sλ(ε) ,x(ε) + O(ε),
close to spheres Sλ(ε) ,x(ε) . More precisely, S (ε) are given by immersions θ(ε) : Sn → Rn+1 , which are
translated graphs,
θ(ε) (ω) := x(ε) + r(ε) (ω)ω,
over spheres, with the graph functions r(ε) of the form r(ε) := λ(ε) + O(ε2 ).
54
Theorem 51. The solutions, S (ε) , corresponding to minima of Rε are asymptotically stable and those
corresponding to saddle points are unstable.
Note that the mean and scalar curvatures of (Rn+1 , e−2ϕε (x) dx2 ) can be expressed in terms of the
conformal factor e−2ϕε (x) as (8.33) below (see and (A.22)) and (see (A.24) and [?])
(8.6)
Rε = ne2ϕε 2∆eucl ϕε − (n − 1)|∇eucl ϕε |2 .
To explain the first result, we mention first that for ε = 0, i.e. the euclidian ambient metric, VPF has
the following properties:
(a) The ε = 0 VPF is invariant with respect to scaling transformations, translations and rotations,
(b) The ε = 0 VPF has the euclidian spheres Sλ,z as static solutions.
For the non-euclidean metric the VPF has still the scaling symmetry, while translational and rotational
symmetries are broken. To be specific we take θ : Sn → Rn+1 , z ∈ Rn+1 . Since
(8.7)
Hε (x)ε=0 = Heucl (x),
the solutions of this equation with ε = 0, are euclidean spheres. On the other hand, we expect that solutions
of the static VPF equation, for ε > 0 and ε = 0 differ by O(ε). This would give θ(ε) (ω) = λω + O(ε) for
some constant λ. The selection of the discrete set zi is a more delicate matter (cf. the nonlinear Schrödingier
equation with an external potential where a similar phenomenon occurs, see [?, ?, ?, ?] and, for a general
discussion, the introduction to [35]).
We will solve (8.5), rather than (8.4). There is an essential difference in treatment of these two equations:
while (8.4) is satisfied by any constant mean curvature surface, (8.5), only by those with the mean curvature
equal to h. This difference is reflected in the spectra of the linearized operators (in the normal direction)
˜ α,0 φ =
for both problems, atRthe sphere Sλz already for = 0. For (8.4), the linearized operator is L
1
n
1
− λ2 (∆ + n)φ + |Sn |λ2 Sn φ, while for (8.4), it is Lα,0 φ = − λ2 (∆ + n)φ, where α = (λ, z). While ω 0 ≡ 1 is
n
an eigenfunction of Lα,0 with the eigenvalue 0, it is an eigenfunction of Lloc
α,0 with the eigenvalue − λ2 , i.e.
the eigenvalue due to the scaling of the surface (see below) gets shifted.
We outline our approach for the problem (1).
Normal graphs over the n− dimensional sphere. We look for solutions of (8.5) as normal graphs over the n−
dimensional unit sphere, Snz , in Rn+1 , centred at centered at z,
θρz (ω) := z + ρ(ω)ω.
(8.8)
We call θρz the shifted normal graph over the sphere. In this representation, the sphere of the radius λ and
centered at z is given by ρ = λ. We call z the center of the surface S, if it is determined by the conditions
Z
(θ(ω) − z) · ω)ω i = 0, i = 1, . . . , n + 1.
(8.9)
Sn
55
These are n+1 equations for the n+1 unknowns z ∈ Rn+1 . According to (8.9), the spheres θλz = θρ=λ,z , λ ∈
R+ , have the centers z, which justifies our interpretation of its solutions as the centre of a surface.
Manifold of spheres. We define the manifold of spheres as
M = {θλz : (λ, z) ∈ R+ × Rn+1 }.
We show that if θ is sufficiently close to M, then it can be written as
θ = θρz , with z s.t ρ ⊥ ω j , j = 1, . . . , n + 1.
(8.10)
Here and in what follows the inner product and orthogonality relation is understood in the sense of L2 (Sn ).
The linearized map. We consider the linearized map Lλ,z,ε = dρ Hε (θρz )ρ=λ on the sphere ρ = λ. (This is
the normal hessian of the functional Vh (θ).) A short contemplation shows that it is a small perturbation of
the operator Lα,0 , where, recall, α = (λ, z), for ε = 0, namely Lα,ε = Lα,0 + O(ε). The latter operator has
especially simple form. Indeed, using the formula for the normal hessian on the sphere, we find
Lα,0 = −
1
(∆ + n),
λ2
(8.11)
where ∆ is the Laplace - Beltrami operator on the sphere, This operator is well understood. Considered on
L2 (Sn ), it is self-adjoint, has purely discrete spectrum, with eigenvalues accumulating at +∞. Its eigenvalues,
their multiplicities and the corresponding eigenfunctions are well-known. In particular, the operator ∆ + n
has
the first, smallest eigenvalue −n which is simple with the eigenfunction ω 0 = 1;
1
the second eigenvalue 0 of multiplicity n + 1 having the eigenfunctions ω , . . . , ω
(8.12)
n+1
;
and the remaining eigenvalues ≥ n + 2.
(8.13)
(8.14)
This result shows that the spectrum of the operator Lα,ε has lots of zero or almost zero eigenvalues.
Hence we cannot use the implicit function theorem to solve the equation (8.5).
The Lyapunov-Schmidt decomposition. We look for the solution of (8.5) in the form (8.8), with
Z
ρ(ω) = λ + φ(ω), where λ = hρi
and
φ = 0.
(8.15)
Sn
R
Here hf i denotes the average, hf i := |S1n | Sn f , of f over the standard n−sphere, Sn and λ is the leading
R
term and represents the sphere centred at 0 of radius λ. The property Sn φ = 0 can be written as φ ⊥ 1
in L2 (Sn ). By (8.16), we have in addition that φ ⊥ ω j , j = 1, . . . , n + 1, which together with the previous
conditions can be written as
φ ⊥ 1,
φ ⊥ ω j , j = 1, . . . , n + 1,
(8.16)
Let A(ρ, z, ε) := Hε (θρz ) − h, where θρz is given in (8.8). Then the equation (8.5) can be rewritten as
A(ρ, z, ε) = 0. Plug the decomposition (8.15) into the latter equation to obtain an equation for φ, λ and z:
A(λ + φ, z, ε) = 0.
(8.17)
56
The explicit form of the map A is given in (8.31) below. We treat this equation by using the LyapunovSchmidt splitting as in [?, 3]. Let P0 be the orthogonal projection onto the eigenspaces of ∆+n corresponding
to the non-positive eigenvalues (i.e. the eigenvalues 0 and −n) and P¯0 = 1 − P0 , the orthogonal projection
onto (Null (∆ + n))⊥ . We decompose the equation (8.17), considered as the equation for φ ∈ Ran P¯0 , λ, and
z, as
P0 A(λ + φ, z, ε) = 0,
P¯0 A(λ + φ, z, ε) = 0.
(8.18)
(8.19)
We consider these two equations for two unknowns φ and α. Now, it is easy to show that the implicit
function theorem is applicable to the second equation, (8.19), as an equation for φ, and gives a unique
solution φ = φα,ε ∈ P¯0 H s , satisfying φα,0 = 0. More precisely, we have
Lemma 52. The equation (8.19) for φ has a unique solution φ = φα,ε ∈ P¯0 H s , satisfying φα,0 = 0.
Proof. We denote F (φ, α, ε) := P¯0 A(λ + φ, z, ε), where α = (λ, z), and use the implicit function theorem
(IFT) for F (φ, α, ) = 0. We go through the checklist for IFT: we claim that
(a) F : P¯0 H 2 × Rn × R → P¯0 L2 and is C 1 in φ and C in α and ;
(b) F (0, α, 0) = 0 ∀ α;
(c) ∂φ F (0, α, 0) is invertible.
Properties (a) and (b) are straightforward (e.g. for (b), write A(α, ε) = A(λ, z, ε), then F (0, α, 0) =
P¯0 A(α, 0) = 0). Since ∂ρ G(α, 0) is self-adjoint and has a purely discrete spectrum and Null ∂ρ A(α, 0) =
Ran P¯0 , we conclude, by the spectral theory (see [36]), that ∂φ F (0, α, 0) = P¯0 ∂ρ A(α, 0) = P¯0 (∆ + n) is
invertible on Ran P¯0 . This gives a unique solution φ = φα,ε of (8.19). By (b), we have that φα,0 = 0.
We plug the solution, φ = φα,ε , to (8.19) into (8.18) to obtain the equation for α
f (α, ε) = 0,
where
f (α, ε) := P0 A(λ + φα,ε , z, ).
(8.20)
This is our reduced equation we want to solve for α. To sum up,
if α(ε) = (zε , λε )
then
(α
(ε)
, φα(ε) ,ε )
solves (8.20), where φ = φα,ε
solves (8.17).
is a unique solution to (8.19),
(8.21)
This implies that θ() := θr() ,z() := z () + r() (ω)ω solves (8.5), and therefore it solves also (8.4), which is
a static equation for (8.3). (All these solutions are parametrized by h > 0, which we do not display in the
notation.) Thus we want to solve (8.20) for α.
Since Lα,0 := ∂ρ A(α, 0) = −λ−2 (∆ + n) is self-adjoint and since P0 is the orthogonal projection onto
the eigenspaces of Lα,0 corresponding to the eigenvalues 0 and −n/λ2 , we have P0 Lα,0 = −(n/λ2 )P1 , where
P1 is the orthogonal projection onto ω 0 = 1. and therefore
P0 Lα,0 φα,0 = 0,
(8.22)
57
This equation and the relations A(α, 0) = 0 and ∂z f (α, 0) = (∂z P0 )G(α, 0) + P0 Lα,0 ∂z φα,0 , imply, ∀ α,
f (α, 0) = 0,
∂z f (α, 0) = 0.
(8.23)
Therefore, we cannot use the implicit function theorem. But (8.20) is a system of n + 2 equations for n + 2
unknowns and we can solve it by other means.
Effective equations. To proceed it is important to observe that the spectral result above implies in
particular that
Ran P0 = span{1, ω 1 , · · · , ω n+1 }.
(8.24)
Hence, since {1, ω 1 , · · · , ω n+1 } are linearly independent, (8.20) is equivalent to the system of n + 2 equations
sj (α, ε) = 0,
j = 0, . . . , n + 1, where sj (α, ε) := hω j A(λ + φα,ε , ε)i,
(8.25)
for n+2 variables α = (λ, z 1 , . . . , z n+1 ). The equation coming from the eigenfuction 1, hA(λ+φα,ε , z, )i = 0,
gives an equation for λ and the equation coming from the eigenfuctions z 1 , . . . , z n+1 give equations for
z 1 , . . . , z n+1 .
We assume for simplicity of notation that ϕε (x) in (8.2) can be written as
ϕε (x) = ϕ0 (x∗ + ε(x − x∗ )).
(8.26)
To simplify notation, we take x∗ = 0, so that ϕε (x) = ϕ0 (x∗ + ε(x − x∗ )) = ϕ0 (εx). Now, expanding λ, z, φ
in ε, we can show that
sj (α, ε) = −
n2
∇0 R(0)ε3 + rem4 ,
2(n + 3)h2 ε3 j
j = 1, . . . , n + 1,
(8.27)
The equation (8.27) implies that for the equations (8.25) to have a solution the origin should be a critical
point of the scalar curvature, as claimed:
∇0 R(0) = 0.
(8.28)
However, the shift z at 0 is still undetermined by these equations. Moreover, we obtain the leading order
correction for the surface ϕ,
φ(ω) =
2h3 (n
n4
1
Ric(0)(ω, ω) −
R(0) ε2 + O(ε3 ).
+ 2)(n − 1)
n+1
(8.29)
This equation gives the leading approximation to the Euclidean sphere.
Going to the fifth order in the expansion one obtains that, if ∇0 ∇0 Rε (0) is non-degenerate, then the
equations sj (α, ε) = 0, j = 1, . . . , n + 1, have a unique solution for z and this solution is of the form
z = z∗ + O(ε) (z1 satisfies the equation of the form ∇0 ∇0 Rε (0)z1 = O(ε4 )). These straightforward, though
somewhat lengthy, calculations can be found in [4]. At the end we find the solutions λ(ε) and z (ε) of (8.20)
or (8.25) and these solutions obey the estimates λ(ε) = nh + O(ε2 ) and z (ε) = O(ε) (or going back to the
original critical point z (ε) = z∗ + O(ε)). By (8.21), we conclude, by (8.21), that the pair
(r(ε) , z () ), where r(ε) := λ(ε) + φα(ε) ,ε ,
(8.30)
solves (8.17). Since ρ, z solving the equation (8.17) is equivalent to z + θρ solving the equation (8.5), this
implies that θ() := θr() ,z() := z () + r() (ω)ω solves (8.5) (and therefore it solves also (8.4)). This finishes
the thumbnail sketch of Theorem 50. MCF Lectures, November 23, 2014
58
Remark. The problem we are dealing with can be rescaled so that the metric becomes fixed, but we are
considering CMC surfaces near spheres of radii converging to 0. In this limit, the asymptotics of the surface
area and of the enclosed volume of geodesic spheres in a Riemannian manifold are well known (see e.g.
[32, 34, 33, 67]) and are expressed in terms of the scalar curvature of the ambient manifold. Hence scalar
curvature pops up also in the expansion of the effective modified area v. This is how the scalar curvature
appears in .
The explicit form of the map A. We show now that the map A(ρ, z, ε) := Hε (θρz ) − h, where
θρz (ω) := z + ρ(ω)ω, has the following explicit form
A(ρ, z, ε) := eϕρz h0 (ρ) − nµ(ρ)−1 (ω − ρ−1 ∇ρ) · ∇z ϕρz − h,
(8.31)
where the parameter h > 0 is not displayed in the notation, µ(ρ) :=
p
1 + ρ−2 |∇ρ|2 , where we denoted
ϕρz (ω) := ϕε (θρz ) = ϕε (z + ρ(ω)ω)
and the map h0 (ρ) is given by
h0 (ρ) := −µ(ρ)−1 {
1
1
n
1
∆ρ − − 3
[ (∇ρ · Hess(ρ)∇ρ) + |∇ρ|2 ]}.
2
2
ρ
ρ
ρ µ(ρ) ρ
(8.32)
∂w
∂ρ
j
k
k
Here we used the following notation: ∇i ρ = ∂u
i and (∇i w)j = ∂ui − Γij wk , where Γij are the Christophel
symbols (here and in what follows the summation over the repeated indices is assumed; see Appendix A),
∆, the Laplace-Beltrami operator on Sn in the standard metric and (Hess)ij = ∇i ∇j .
To derive (8.31) - (8.32), we recall that the mean curvature for a hypersurface S immersed in (Rn+1 , e−2ϕε dx2 ),
is given by (see (A.22))
Hε (x) = eϕε (H0 − n∇ν ϕε ),
(8.33)
where H0 (x) = Heucl (x), ν is the euclidean unit normal vector field on S and ∇ν f = νf is the the Euclidean
directional derivative in the direction ν. Note that ν is related to the conformal unit normal vector field,
νε (x), on S by νε (x) = eϕ ν(x). We plug the expression (8.33) into the definition A(ρ, z, ε) := Hε (θρz ) − h,
to obtain
A(ρ, z, ε) = eϕρz H0 (θρz ) − n∇ν ϕρz − h.
(8.34)
Now, we express the first two terms on the r.h.s. in (8.34) explicitly in terms of ρ and z. Using the relations
H0 (θρ + z) = H0 (θρ ), and ν(θρ + z) = ν(θρ ), and using Proposition 59 of Appendix A.8 which gives an
explicit expression of H0 (θρ ) in terms of ρ, (or the level set representation, as in [3, 56], Appendix A.1) and
the relation (A.38), ν = (1 + ρ−2 |∇ρ|2 )−1/2 (ω − ρ−1 ∇ρ), we obtain the expression (8.31) for the map A.
8.1
Variational formulation
Recall that surfaces of constant mean curvature (CMC) are critical points of the area functional V (ψ) on
space of immersions with the given enclosed volume Vencl (ψ). Furthermore, by the Lagrange multiplier
59
theorem, these are critical points of the functional Vh (ψ) := V (ψ) − hVencl (ψ), where h is determined by
c = Vencl (ψ) and vice versa (see Proposition 46).
Now, let ψα := θα (ω) + φα,ε (ω)ω, where θα (ω) := z + λω and φα,ε , α = (λ, z), is a unique solution to
(8.19). We define the function
v(α) := Vh (ψα ).
(8.35)
The expansion v(α) in ε is done similarly to (8.27), with the leading non-trivial term proportional to the
scalar curvature of the ambient space (see [32, 33, 67, 61] and the remark above). We have
Theorem 53. ∂α v(α) = 0 ⇐⇒ dVh (ψα ) = 0 ⇐⇒ f (α) = 0.
Proof. We compute ∂α v(α) = dVh (ψα )∂α ψα . Hence, dVh (ψα ) = 0 =⇒ ∂α v(α) = 0. Next, by the definition,
dVh (ψα )ξ = hVh0 (ψα ), ξi, of the gradient Vh0 (ψ) of Vh (ψ), we have Vh0 (ψ) = Hε (ψ) − h. (This is the place
where we used the modification of the Lyapunov-Schmidt equations.) Hence, the P¯0 −equation (8.19) implies
that P¯0 Vh0 (ψα ) = 0 and therefore
P0 Vh0 (ψα ) = Vh0 (ψα ).
(8.36)
Note that P0 is the projection
Tθα M0 , where M0 := {θα }. Let P be the projection onto Tψα M,
P onto
where M := {ψα }. Then P h =
g ij h∂αi ψα , hi∂αj ψα where gij := h∂αi ψα , ∂αj ψα i. Using the basis {∂αj ψα }
and the relation ∂α v(α) = hVh0 (ψα ), ∂α ψα i, we find
X
P Vh0 (ψα ) =
g ij ∂αi v(α)∂αj ψα .
(8.37)
This gives ∂α v(α) = 0 =⇒ P Vh (ψα ) = 0. Using the last relation, together with P0 Vh0 (ψ) = Vh0 (ψ), we find
Vh0 (ψ) = P0 Vh0 (ψα ) = P Vh0 (ψ)+(P0 −P )Vh0 (ψα ) = (P0 −P )Vh0 (ψα ), which gives kVh0 (ψα )k ≤ k(P0 −P )Vh0 (ψα )k.
Since, k(P0 − P )k < 1, for ε sufficiently small, we have Vh0 (ψα ) = 0 as required. Finally, the equations
f (α) = P0 Vh0 (ψα ) and the relation P0 Vh0 (ψ) = Vh0 (ψ) imply f (α) = 0 ⇐⇒ dVh (ψα ) = 0.
Stability.
We have
Theorem 54. The surface ψα∗ is stable (in the sense of geometric analysis) ⇐⇒ the hessian of v(α) at α∗
is positive definite (i.e. α∗ is a minimum of v(α)).
Proof. Differentiating the equation the relation ∂α v(α) = hVh0 (ψα ), ∂α ψα i and using that ∂αj Vh0 (ψα ) =
Vh00 (ψα )∂αj ψα , where Vh00 (ψ) = dVh0 (ψ) is the hessian of Vh (ψ), we find ∂αi ∂αj v(α) = hVh00 (ψα )∂αj ψα , ∂αi ψα i+
hVh0 (ψα ), ∂αi ∂αj ψα i. At a critical point α∗ of v(α), this gives
∂αi ∂αj v(α∗ ) = hVh00 (ψα∗ )∂αj ψα , ∂αi ψα i.
(8.38)
Since the spectrum of the operator Vh00 (ψα∗ ) restricted to (T M)⊥ is bounded below by n + 2 − O(ε) > 0
(see (8.12) - (8.14)), the smallest eigenvalues of Vh00 (ψα∗ ) are coming from its restriction to T M and these
eigenvalues determine the stability of ψα∗ . By the equation (8.38), these eigenvalues are given by the
eigenvalues of the hessian v 00 (α∗ ) of v(α).
60
0
Dynamics. Recall the definitions θα (ω) := z + λω and α = (λ, z) and let gij
:= h∂αi θα , ∂αj θα i. We denote
˙
by ψ = dψ/dt the total derivative in t. We have
Theorem 55. For δ = (n + 2)/2, we have
kα˙ k −
X
g0ki ∂αi v(α)k . ε2 e−δt .
(8.39)
N
Proof. We consider the VPF equation ψ˙ = Vh0 (ψ) (or ψ˙ N = Vh0 (ψ)). Write ψα (ω, t) = θα (ω) + Φ(ω, t). We
˙ This gives ∂α θα α˙ + Φ˙ = V 0 (θα + Φ). Applying the projection P¯0 to this equation
compute ψ˙ = ∂α θα α˙ + Φ.
h
and using that P¯0 ∂α θα α˙ = 0, we obtain P¯0 Φ˙ = P¯0 Vh0 (θα + Φ). Since P¯0 Φ˙ = Φ˙ + P˙0 Φ, this implies
Φ˙ = P¯0 Vh0 (θα + Φ) − P˙0 Φ.
(8.40)
Let Φ ≡ Φα is aP
unique solution to the previous equation. On the other hand, multiplying the same equation
0 j
˙ = h∂αi θα , V 0 (θα + Φ)i. Multiplying this by the inverse matrix g ij and
by ∂αi θα gives
gij
α˙ + h∂αi θα , Φi
0
h
˙ = −h∂αi ∂αj θα α˙ j , Φi, we find
using that h∂αi θα , Φi
X
X
α˙ k −
g0ki hΦ
˙j =
g0ki h∂αi θα , Vh0 (θα + Φα )i,
ij α
where hΦ
ij := h∂αi ∂αj θα , Φi. Here hij is the matrix of the second fundamental form of the surface M0 := {θα }.
Next, by the above, we have ∂α v(α) = hVh0 (ψα ), ∂α ψα i = hVh0 (ψα ), ∂α θα i + hVh0 (ψα ), ∂α Φα i. The last two
equations give
X
X
X
g0ki h∂αi Φα , Vh0 (θα + Φα )i,
α˙ k −
g0ki hij α˙ j −
g0ki ∂αi v(α) = −
which implies (one can probably do a bit better, if the evolution is slow and so Vh0 (ψ) is small)
X
(8.41)
kα˙ k −
g0ki ∂αi v(α)k . kΦk.
The norm on the r.h.s. is controlled using the equation for Φ above and the Lyapunov functional technique
of Section 5.6 with the Lyapunov functionals Λk (φ) := 21 hφ, (−∆ − n)k φi. The key point here is that, due
to (8.12) - (8.14) and (8.16), these functionals satisfy the inequalities (see [3])
Λk (φ) ≥ (n + 2)Λk−1 (φ) and Ckφk2H k ≥ Λk (φ) ≥ ckφk2H k ,
(8.42)
for some C > c > 0. Using these relations and the equation (8.40) for φ, we derive the differential inequalities
for Λk (φ), which imply that Λk (φ) . ε2 e−δt , for δ = (n + 2)/2, which, together with (8.41) and (8.42), gives,
in turn, (8.39).
∗∗∗∗∗∗∗∗∗∗∗∗∗
∗∗∗∗∗∗∗∗∗∗∗
∗∗∗∗∗∗∗∗∗∗∗
9
10
61
Ricci flow (to do)
Presentation topics
1) Local existence for the MCF (Huisken and Polden, Geometric evolution equations for hypersurfaces, Sect
7)
2) Monotonicity formulae I (Tobias Holck Colding, William P. Minicozzi II, Monotonicity - analytic and
geometric implications, arXiv:1205.6768)
3) Monotonicity formulae II (Andrews, Ben Noncollapsing in mean-convex mean curvature flow. Geom.
Topol. 16(2012), no. 3, 14131418)
4) Compactness theorem for surfaces (Langer, Compactness theorem for surfaces with Lp-bounded second fundamental form, Math. Ann. 270 (1985), 223-234)
5) The maximal time of existence of MCF (Huisken, Flow by mean curvature of convex surfaces into
spheres, Journal of Differential Geometry 20 (1984) 237-266, Theorem 8.1 and Corollary 9.4)
6) A bound on |∇H| (Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of
Differential Geometry 20 (1984) 237-266, Theorem 6.1)
7) Huisken’s surface collapsing theorem (Huisken, Flow by mean curvature of convex surfaces into
spheres, Journal of Differential Geometry 20 (1984) 237-266)
8) Mean curvature evolution of entire graphs (Ecker and Huisken, Mean curvature evolution of entire
graphs, Annals of Mathematics, 130 (1989), 453-471 )
9) Propagation of convexity (the book: Carlo Mantegazza, Lecture Notes on Mean Curvature Flow, 3.5.
Convexity Invariance and most of Chapter 3 for the background, and Huisken, Flow by mean curvature of
convex surfaces into spheres, Journal of Differential Geometry 20 (1984) 237-266, Sect 4).
10) Classification theorem for self-similar surfaces (G. Huisken, Asymptotic behavior for singularities
of the mean curvature flow, J. Differential Geom., 31(1):285–299, 1990, Thm 4.1, or Colding and Minicozzi
(Generic mean curvature flow I; generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755833 )
11) Rigidity of spheres and cylinders (Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi II,
Brian White, The round sphere minimizes entropy among closed self-shrinkers, arXiv:1205.2043)
12) Monotonicity formulae III (Ecker, Knopf, Ni and Topping, Local monotonicity and mean value
formulas for evolving Riemann manifolds, arXiv:math.DG/0608470 v1)
13) Planar network flow (Mazzeo and Saez, Self-similar solutions for the planar network flow, arXiv:0704.3113v1,
Mantegazza et al, Evolution by curvature of networks of curves in the plane, Progress in nonlinear diff eqs
and their applications, vol 95, 95-109, 2004, Schnuker et al, arXiv, 2007)
14) Rate of convergence of the mean curvature flow (N. Sesum, Rate of convergence of the mean curvature
flow, Comm on Pure and Appl Math, LXI, 0464-0485 (2008))
15) Translating solitons (Clutterbuck et al, Stability of translating solutions to mean curvature flow,
arXiv:math/0509372v1 and/or Gui, Jian and Ju, Properties of of translating solitons to mean curvature
flow, arXiv:0909.5021v1 )
62
16) Volume-preserving mean curvature flow of rotationally symmetric surfaces (M. Athanassenas, Volumepreserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv., 72(1), pp. 5266,
1997).
17) Stability of noncompact CMC surfaces (de Silveira, Stability of complete noncompact surfaces with
CMC, Math Ann 277, 629-638 (1987)).
18) Stable periodic constant mean curvature surfaces (Ros, Antonio Stable periodic constant mean
curvature surfaces and mesoscopic phase separation. Interfaces Free Bound. 9 (2007), no. 3, 355365, can be
found on-line).
19) The Minkowski formula (Ch-Ch Hsiung, Some integral formulas for closed hypersurfaces, Math.
Scand. 2 (1954), 286 - 294).
===================
20) Anisotropic mean curvature flow (see B. Andrews, Volume-preserving anisortropic MCF, Indiana
Univ Math J 50, No 2, 2001, 763.)
21) Gray, A., The volume of a small geodesic ball of a Riemannian manifold, Michigan Math. J., 20
(1973), 329-344.
22) Monotonicity formulae IV (Tobias Holck Colding, New monotonicity formulas for Ricci curvature
and applications; I, arXiv:1111.4715 )
23) Allen-Cahn to Brakke’s MCF (Imanen, Convergence , J Diff Geom, 35 (1993), 417-461)
24) Ricci solitons, their entropy and reduced distance (Ivey, Ricci solitons , Diff Geom and its applications
3 (1993) 301-307; Huai-Dong Cao, Recent progress on Ricci solitons, ArXiv;0908.2006v1, 2009; Feldman,
Ilmanen, Ni, Entropy and reduced distance for Ricci expanders).
25) Kaehler-Ricci solitons (Feldman, Ilmanen, Knopf, Rotationally symmetric shrinking and expanding
Kaehler-Ricci solitons, Journal Differential Geometry 65 (2003) 169-209,
26) Neckpinching under the Ricci flow (Angenent and Knopf, Precise asymptotics of the Ricci flow
neckpinch, Communications in Analysis and Geometry vol 15, N 4, 773-844, 2007)
27) UNIQUENESS OF BLOWUPS AND LOJASIEWICZ INEQUALITIES (TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II, UNIQUENESS OF BLOWUPS AND LOJASIEWICZ INEQUALITIES, http://arxiv.org/pdf/1312.4046.pdf)
A
Elements of Theory of Surfaces
In this appendix we sketch some results from the theory of surfaces in Rn+1 . For detailed expositions see
[57, 14, 21].
By a hypersurface S in Rn+1 , we often mean an n dimensional surface element, which is an equivalence
class of parameterized surface elements, i.e. maps ψ : U → Rn+1 , where U ⊂ Rn is open, s.t. dψ(u) is oneto-one (i.e. of the maximal rank) for every u ∈ U . Such maps are called (local) immersions. Here dψ(u) is
the linear map from Rn to Rn+1 , defined as dψ(u)ξ = ∂s |s=0 ψ(σs ), where σs is a curve in U so that σs=0 = u
n+1
s
, ψ 0 : U 0 → Rn+1 ,
and ∂σ
∂s = ξ. The equivalence relation is given as follows: two immersions, ψ : U → R
63
are said to be equivalent, if there is a diffeomorphism, ϕ : U 0 → U , s.t. ψ 0 = ψ ◦ ϕ.
We call ψ a parametrization of S. We are interested in properties of S, which are independent of a
particular parametrization, i.e. depend only on the equivalence class of parametrizations.
The tangent space, Tx S, to S at x is defined as the vector space of velocities at x of all curves in S
s
passing through x, i.e. Tx S := {η ∈ Rn+1 : ∃γs : [−, ] → S, s. t. γs=0 = x and ∂γ
∂s = η}. Let γs be as in
−1
the definition and σs := ψ (γs ). Then σs is a curve in U through the point u = ψ −1 (x) and η = dψ(u)ξ,
n
s
where ξ := ∂σ
∂s . Thus, we have Tx S = dψ(u)(R ) for x = ψ(u).
The inner product on Tx S is defined in the usual way: g(v, w) = hv, wi , ∀v, w ∈ Tx S. Here h·, ·i is the
inner product in Rn+1 . The form g(v, w) is called the metric on S or the first fundamental form of S.
Often, a surface S will be given either as a level set of some function ϕ : Rn+1 → R, i.e. S = ϕ−1 (0) =
{x ∈ Rn+1 : ϕ(x) = 0}, or as a graph of some function f : U → R. In the latter case, it is a zero level set of
the function ϕ(x) = xn+1 − f (u), x = (u, xn+1 ), and it has the parametrization ψ(u) := (u, f (u)).
A.1
Mean curvature
Let ν(x) be the unit outward normal vector to S at x. The map ν : S → S n given by ν : x → ν(x) is
called the Gauss map and the map Wx := dν(x) : Tx S → Tν(x) S n is called the Weingarten map or the shape
s
operator. Here dν(x)η = ∂s |s=0 ν(γs ), where γs is a curve on S so that γs=0 = x and ∂γ
∂s = η. We have the
following theorem.
Theorem 56. Ran(Wx ) ⊂ Tx S and Wx is self-adjoint, i.e. g(Wx ξ, η) = g(ξ, Wx η).
Proof. The statement Ran(Wx ) ⊂ Tx S is trivial, since as a little contemplation shows, Tν(x) S n = Tx S (both
subspaces are defined by the same normal vector ν(x)). It also follows from the elementary, but useful,
computation
1
hWx ξ, ν(x)i = h∂s |s=0 ν(γs ), ν(x)i = ∂s |s=0 hν(γs ), ν(γs )i = 0,
2
where γs is as above. This implies that Wx ξ ∈ Tx S.
Let ϕ = ϕst be a parametrization of a two-dimensional surface in S such that ϕst |s=t=0 = x, ∂s |s=t=0 ϕst =
ξ and ∂t |s=t=0 ϕst = η. Then
hξ, Wx ηi = h∂s |s=t=0 ϕ, ∂t |s=t=0 ν(ϕ)i = ∂t |s=t=0 h∂s ϕ, ν(ϕ)i − h∂t ∂s |s=t=0 ϕ, ν(x)i .
D 2
E
∂ ϕ
Since h∂s ϕ, ν(ϕ)i = 0, this implies hξ, Wx ηi = − ∂s∂t
(0, 0), ν(x) . Similarly we have
hWx ξ, ηi = −
∂2ϕ
(0, 0), ν(x) ,
∂s∂t
which implies that hξ, Wx ηi = hWx ξ, ηi .
The form hv, Wx wi is called the second fundamental form of S.
64
The principal curvatures are all the eigenvalues of the Weingarten map Wx . The mean curvature H(x)
is the trace of Wx , which is the sum of all principal curvatures,
H(x) = Tr Wx .
(A.1)
The Gaussian curvature is the determinant of Wx , which is the product of the principal curvatures.
If S is a level set of some function ϕ : Rn+1 → R, i.e. S = ϕ−1 (0) = {x ∈ Rn+1 : ϕ(x) = 0},
∇ϕ(x)
then ν(x) = |∇ϕ(x)|
. Note that ν(x) is defined on the entire Rn+1 and therefore we can introduce the
matrix ATx = (∂xi ν j (x)). The associated linear map has the following properties: a) ATx ν = 0 and b)
Wx = Ax |Tx S . Indeed, since 0 = ∂xi ν j ν j = 2ν j ∂xi ν j , we have ATx ν = 0. Furthermore, let γs be a
curve through a point x, with the initial velocity ξ, i.e. γs (0) = x, and with ∂s γs |s=0 = ξ. We have
∂γ i
dν(x)ξ = ∂s |s=0 ν(γs ) = ∂xi ν(x) ∂ss |s=0 = ∂xi νξ i = Ax ξ. This implies that Wx = Ax |Tx S . The properties a)
and b) yield that Tr(Wx ) = Tr(Ax ) = ∂xi ν i (x) = div ν(x), which gives
∇ϕ(x)
H(x) = div ν(x) = div
.
(A.2)
|∇ϕ(x)|
If S is a graph of some function f : U → R, then it is a zero level set of the function ϕ(x) = xn+1 −
√ (u),1)
and
f (u), x = (u, xn+1 ), and therefore by the formula above ν(x)|x=ψ(u) = (−∇f
2
1+|∇f |
H(x) = − div
A.2
∇f
p
1 + |∇f |2
.
(A.3)
Matrix representation of the first and second fundamental forms
We denote by T S the collection of all tangent planes, the tangent bundle. A vector field V is a map
∂ψ
V : S → T S assigning to every x ∈ S a tangent vector Vx ∈ Tx S. Given an immersion ψ : U → S, { ∂u
i (u)}
∂ψ
is a basis in Tx S, where x = ψ(u). Varying u ∈ U , we obtain the basis, { ∂u
i }, for vectors fields on S (if
∂
the immersion ψ is fixed, one writes this basis as { ∂u
i }). This allows to write vector fields in the coordinate
i ∂ψ
form as v = v ∂ui . Plug the coordinate form of v and w into g(v, w) to obtain g(v, w) = gij v i wj , where
∂ψ ∂ψ
.
gij :=
,
∂ui ∂uj
∂ψ
The matrix (tensor) {gij } determines the first fundamental form of S in the basis { ∂u
i }. If S is a graph of
∂f ∂f
some function f : U → R, i.e. S can be parameterized as ψ(u) = (u, f (u)). Then gij (u) = δij + ∂u
i ∂uj .
Notation. {g ij } denotes the inverse matrix. The summation is understood over repeated indices. The
indices are raised and lowered by applying g ik or gik as in bij = g ik bkj . Unless we are dealing with the
Euclidian metric, one of the indices are upper and the other lower. (For the Euclidian metric, δij , the
position of the indices is immaterial.) The quadratic form hWx v, wi , where v, w ∈ Tx S, is called the second
fundamental form of S. We have
65
Lemma 57. For a parametrization ψ of S, consider the matrix elements of Wx in the basis
bij := h
∂ψ
∂ui ,
∂ψ
∂ψ
, Wx , j i.
i
∂u
∂u
(A.4)
Then
bij = −h
∂2ψ
, νi
∂ui ∂uj
and
H = g ij bji .
Proof. In the proof of Theorem 56, we have shown that hξ, Wx ηi = −
D
(A.5)
E
∂2ϕ
∂s∂t s=t=0 , ν(x)
, where ϕ = ϕst
is a parametrization of a two-dimensional surface in S such that ϕst |s=t=0 = x, ∂s |s=t=0 ϕst = ξ and
∂t |s=t=0 ϕst = η. Now take ϕst = ψ(u + sei + tej ), where ei are the co-ordinate unit vectors in U . Then
∂2ψ
∂2ϕ
∂s∂t s=t=0 = ∂ui ∂uj , which proves the first equality. To show the second equality in (A.5), we write H(x) =
P
∂ψ
Tr(Wx ) =
hei , Wx ei i, where {ei } is an orthonormal basis of Tx S. Let ei = µij ψj , where ψj = ∂u
j . Let
U = (µij ). Then
hei , Wx ei i = µik µil hψk , Wx ψl i = µik µil bkl = Tr(U BU T ) = Tr(U T U B),
where B = (bji ) . We have δij = hei , ej i = µik µjl gkl = (U GU T )ij , where G = (gij ). Hence U GU T = I and
therefore G−1 = U T U . So
H = Tr(G−1 B)
(A.6)
and therefore the second equation in (A.5) follows.
∂ψ
∂ψ
By (A.4) and the general formula v = g ij h ∂u
i , vi ∂uj , we have the following useful expression
Wx
∂ψ
∂ψ
= g ij bjk k .
i
∂u
∂u
(A.7)
n
n
ˆ
by the immersion ψ(u) = Rψ(u),
of radius R in Rn+1 . We can define SR
Examples. 1) The n−sphere SR
n
n
ˆ
ˆ
where ψ(u) is the immersion for the standard n−sphere S = S1 . Then ν(ψ(u)) = ψ(u) and Lemma 57
2 ˆ
ψ
ˆ = Rh ∂ ψˆ , ∂ ψˆ i = R−1 gij = Rg stand , where gij and g stand are the metrics on S n
gives bij = −Rh ∂u∂i ∂u
, ψi
ij
ij
R
∂ui ∂uj
j
and Sn = S1n , respectively. This in particular implies H = g ij bji = nR−1 .
n−1
n
n
2) The n−cylinder CR
= SR
× R of radius R in Rn+1 . We can define CR
by the immersion φ(u, x) =
n−1
ˆ
ˆ
(Rψ(u), x), where ψ(u) is the immersion for the standard (n − 1)−sphere S
= S1n−1 . Then ν(φ(u, x)) =
2 ˆ
ˆ ∂ψ
ˆ
∂
ψ
∂
ψ
−1
stand
stand
ˆ
ˆ = Rh ,
gij = Rgij
, where gij and gij
(ψ(u),
0) and Lemma 57 gives bij = −Rh ∂ui ∂uj , ψi
∂ui ∂uj i = R
n−1
and Sn−1 = S1n−1 , respectively, for i, j = 1, . . . , n−1, and bnn = 0. This in particular
ij
implies H = g bji = (n − 1)R−1 .
A.3
Integration
on S. Indeed, first, we define the integrals
RThe first fundamental form allows us to define Rthe integration
R
√
√
h over an immersion patche, ψ : U → S, as ψ h := U h ◦ ψ gdn u, where g := det(gij ). ( gdn u is the
ψj
66
∂ψ
infinitesimal volume spanned byR the basis
∂uj .) Then, we extend to the entire surface by using a
R
P vectors
partition of unity, say, {χj }, as S h := j ψj χj h. It is easy to check that this expression is independent
of the choice of partition of unity, {χj }, or local immersions, ψj .
If S is locally a graph, S = graph
f , of some function f : U → R, so that one can take the immersion
p
√
ψ(u) = (u, f (u)), then gdn u = 1 + |∇f |2 dn u.
Show that V (ψ) is independent of the parametrization.
A.4
Connections
The connection, ∇, is probably a single, most important notion of Differential Geometry. It generalizes the
map
n+1
V → ∇R
V
from Rn+1 to manifolds (in our case surfaces). Here V is a vector field on Rn+1 , i.e. V : Rn+1 → Rn+1 ,
n+1
n+1
and ∇R
is the directional derivative which is defined as ∇R
: T (x) 7→ ∂s |s=0 T (γs ), where γs=0 = x,
V
V
n+1
∂γs
|
=
V
and
T
is
either
a
function
or
a
vector
field.
(We
will
not use tensors.) Note that ∇R
T =
s=0
V
∂s
P i ∂T
Rn+1
W (x) ∈
/ Tx S, i.e. “ does not belong to S
i V ∂ui . However, if W is a vector field on S, in general ∇V
”. This suggests the following definition.
If f is a function on S, we define ∇V f (x) = ∇R
V
n+1
W (x) into Tx S,
as the projection of ∇R
V
n+1
f (x). If W is a vector field on S, we define ∇V W (x)
∇V W (x) := (∇R
V
n+1
W )T (x).
(A.8)
∇V is called the covariant derivative on S ⊂ Rn+1 . So in the latter case the connection ∇ : V 7→ ∇V is a
linear map from space V(S), of vector fields on S, to the space of first order differential operators on V(S).
The covariant derivative, ∇V , has the following properties:
• ∇V is linear, i.e. ∇V (αW + βZ) = α∇V W + β∇V Z;
• ∇f V +gW = f ∇V + g∇W ;
• ∇V (f W ) = f ∇V W + W ∇V f (Leibnitz rule).
Homework: show these properties hold.
∇ is called a Levi-Civita or Riemann connection if
∇V g(W, Z) = g(∇V W, Z) + g(W, ∇V Z)
(A.9)
and is called symmetric if
∇V W − ∇W V = [V, W ].
Homework: Show that our connection is a Levi-Civita symmetric connection.
Claim: for a Levi-Civita symmetric connection, one has
hZ, ∇Y Xi =
1
{X hY, Zi + Y hZ, Xi − Z hX, Y i − h[X, Z], Y i − h[Y, Z], Xi − h[X, Y ], Zi}.
2
(A.10)
67
Thus ∇ depends only on the first fundamental form gij . To prove this relation we add the permutations of
X hY, Zi = h∇X Y, Zi + hY, ∇X Zi .
Consider a vector field V on S, i.e. a map V : S → T S. So if γs is a path in S s.t. γs |s=0 = x and
∂s γs |s=0 = V (x), then
∂
|s=0 f (γs ).
∂s
V f (x) = ∇V f (x) =
∂σ
Let σ : U → S ⊂ Rn+1 be a local parametrization of S. Then { ∂u
i } is a basis in Tσ(u) S (generally not
−1
orthonormal). We can write V (x) = ∂s γs |s=0 = ∂s σ ◦ σ ◦ γs |s=0 . By the chain rule we have V (x) =
−1
P ∂σ ∂ −1
∂σ
i
◦ γs )i |s=0 . Hence we can write V = V i ∂u
= ∂((σ ∂s◦γs )i ) |s=0 .
i , where V
i ∂ui ∂s (σ
A vector field V is defined as a map V : S → T S, but is also identified with an operator V f = ∇V f .
Let f : S → R be a function on S. Then
V f (x)
=
=
∂
◦ σ −1 ◦ γs
∂s |s=0 f ◦ σ
∂(f ◦σ) ∂((σ −1 ◦γs )i )
|s=0 ,
∂ui
∂s
which by the above relations can be rewritten as
V f (x) = V i
∂f
.
∂ui
∂
We write V = V i ∂u
i . We think of {∂ui } as a basis in T S ({(∂ui )x }, as a basis in Tx S). We think of ∂ui are
∂σ
either operators acting on functions on S or as vectors ∂ui σ. (Note that the vector ∂u
i has components δij
∂
∂σ
∂
in the basis { ∂uj } and therefore the operator associated with it is δij ∂uj = ∂ui .)
Since {∂ui } is a basis, then ∇∂ui ∂uj can be expanded in it: ∇∂ui ∂uj = Γkij ∂uk , for some coefficients Γkij ,
called the Christofel symbols. If we let X = ∂ui , Y = ∂uj and Z = ∂uk in (A.10), then we find
Γkij =
1 kl ∂glj
∂gli
∂gij
g ( i +
−
).
2
∂u
∂uj
∂ul
This can be also shown directly.
Since ∇V = ∇V i ∂ui = V i ∇∂ui and ∇∂ui W j =
∇V W
∂
j
∂ui W ,
we have
= ∇V i ∂ui (W j ∂uj )
= V i ∇∂ui (W j ∂uj )
= V i [(∇∂ui W j )∂uj + W j ∇∂ui ∂uj ]
= V i (∂ui W j ∂uj + W j Γkij ∂uk ),
which gives ∇V W = V i ∇i W k ∂uk , or (∇V W )k = V i ∇i W k , where
∇i W k ≡ ∇∂ui W k :=
∂W k
+ Γkij W j .
∂ui
A.5
68
Various differential operators
We give definitions of various differential operators used in geometry.
(1) Divergence div V = Tr ∇V , where ∇V is the map W 7→ ∇W V .
(2) Gradient. The vector field grad(f ) is defined by g(grad(f ), W ) = ∇W f ∀ vector field W , where ∇f :
V 7→ ∇V f is considered as a functional on vector fields.
(3) Laplace-Beltrami ∆ := div grad.
(4) Hessian Hess(f ) := ∇2 f . Explicitly, ∇2 f (V, W ) = (V W − ∇V W )f .
Here ∇f is viewed as (0, 1) tensor and ∇V is defined on (0, 1) tensors T : T S → R as (∇V T )(W ) =
∇V (T (W )) − T (∇V W ).
Recall the notation g = det(gij ). In local coordinates, these differential operators have the following
expressions:
1
√
div V = ∇i V i = √ ∂ui ( gV j ),
g
∂f
(grad(f ))i = ∇i f = g ij j ,
∂u
∂2f
∂f
(Hess(f ))ij = ∇i ∇j f =
− Γkij k ,
∂ui ∂uj
∂u
1
∂f
√
∆f = ∇i ∇i f = √ ∂ui ( gg ij j ).
g
∂u
Exercise:
(A.11)
(A.12)
(A.13)
(A.14)
(1) Show (A.11)- (A.14); (2) Show that grad = −(div)∗ and ∆∗ = ∆.
Proof of (A.11)- (A.14).
We have, for an orthonormal basis {ei },
div V = h∇ei V, ei i = g ij ∇∂ui V, ∂uj
= g ij ∇i V k h∂uk , ∂uj i = g ij gkj ∇i V k ,
which gives
div V = ∇i V i
To show div V =
(1.11))
√
√1 ∂ui ( gV j ),
g
we use the representation det(gij ) = eTr ln(gij ) to obtain the formula (cf.
∂g
∂gij
= gg ij k .
∂uk
∂u
√
∂gij
1 ml
√1 ∂uk g = 1 ∂uk g = 1 g ij
. On the other hand, Γm
(∂um gkl
mk = 2 g
g
2g
2
∂uk
√
1 ml
1
m
m
that Γmk = 2 g ∂uk gml and therefore √g ∂uk g = Γmk . So we have
From (A.16), we have
∂ul gmk ). It follows
div V =
(A.15)
∂V i
∂V i
1
1 ∂ √ i
√
i
+ Γm
+ √ ∂ui gV i = √
( gV ).
mi V =
i
∂u
∂ui
g
g ∂ui
(A.16)
+ ∂uk gml −
69
which gives (A.11).
If grad(f ) = (grad(f ))i ∂ui , then
which gives (A.12).
Since ∇i W j ≡ ∇∂ui W j :=
∂W j
∂ui
∂f
∂uj
= ∇∂uj f = (grad(f ))i ∂ui , ∂uj = (grad(f ))i h∂ui , ∂uj i = gij (grad(f ))i ,
+ Γjik W k , we have (Hess(f ))ji =
Furthermore, (Hess(f ))ij := ∇2 f (∂ui , ∂uj ) =
∂2f
∂ui ∂uj
∂
jk ∂f
∂ui g ∂uk
∂f
+ Γjik g k` ∂u
`.
− (∇∂ui ∂uj )f , which gives (A.13).
∂f
Finally combining the previous results, we obtain ∆f = div grad(f ) = div(g ij ∂u
j ∂ui ) =
√
∂f
√1 ∂ui ( gg ij
g
∂uj ).
Proposition 58. Let σ : U → S ⊂ Rn+1 be a local parametrization of S. Then
∆S σ = −Hν.
(A.17)
Proof. We will prove it at a point x0 ∈ S. Since (A.17) is invariant under translations and rotations
(show
T
this), we can translate and rotate S so that x0 becomes a critical point in the sense that S W = graph f
∂σ ∂σ ∂f ∂f
√(−∇f,1) 2 and
and grad(f )(x0 ) = 0. Now σ(u) = (u, f (u)) and gij = ∂u
= δij + ∂u
i , ∂uj
i ∂uj . Then ν =
1+|∇f |
ν|x=x0 = (0, 1). Note that
∆S f
∂f
= ∇i (grad(f ))i = ∇i g ij ∂u
j
∂
ij ∂f
m ij ∂f
= ∂ui (g ∂uj ) + Γmi g ∂uj
∂2f
∂g ij ∂f
m ij ∂f
= g ij ∂u
i uj + ∂ui ∂uj + Γmi g
∂uj .
2
∂ f
Since gij |x=x0 = δij and therefore g ij |x=x0 = δij , we have ∆S f |x=x0 = g ij ∂u
i uj |x=x0 =
Hence
X ∂2f
n+1
Rn+1
∆S σ|x=x0 = (0,
f |x=x0 = ν|x=x0 ∆R f |x=x0 .
2 )|x=x0 = (0, 1)∆
i
i ∂u
n+1
But H(x0 ) = − divR
Rn+1
(√ ∇
f
1+∇Rn+1 f 2
)|x=x0 = − divR
n+1
∇R
n+1
f |x=x0 = −∆R
n+1
∂2f
i ∂ui 2 |x=x0 .
P
f |x=x0 , this completes the
proof of the proposition.
A.6
Submanifolds
Recall that the Weingarten map is defined as Wx V = dν(x)V , which can be written in terms of the Euclidean
n+1
connection ∇R
as
W x V = ∇R
V
n+1
ν(x).
(A.18)
For the second fundamental form, we have consequently that for ξ, η ∈ T Rn+1 ,
Wx (ξ, η) = h∇ξR
n+1
ν(x), ηi = −hν(x), ∇R
ξ
n+1
ηi.
If S is a hypersurface in an (n + 1)− dimensional Riemannian manifold (M, s), rather than in the Euclidean
space Rn+1 , we define the Weingarten map by
W x ξ = ∇M
ξ ν(x),
(A.19)
70
where ν is the outward normal vector field on S in the metric s and ∇M is a (Levi-Civita symmetric)
connection on M , the second fundamental form by
M
Wx (ξ, η) = h∇M
ξ ν(x), ηis = −hν(x), ∇ξ ηis ,
(A.20)
where we used the relation (A.9) for Levi-Civita connections, and the mean curvature, still by (A.1).
As an example, we consider the Euclidean space Rn+1 , with the conformally flat metric σ 2 dx2 , where
σ > 0. Let σ = e−ϕ and denote ∇M by ∇σ . Then we have
n+1
∇σξ η = ∇R
ξ
η − (ξϕ)η − (ηϕ)ξ + hξ, ηiσ grad ϕ,
where ξf = ∇ξ f is the the Euclidean directional derivative (application of the vector field to a function).
Moreover, let ν and ν σ be the outward normal vector fields on S in the euclidean metric and in the metric s.
n+1
Using that hξ, νiσ = σ 2 hξ, νi = 0 for ∀ξ ∈ T S, we have ν σ = σ −1 ν and Wxσ ξ = ∇σξ ν σ (x) = ∇R
(σ −1 ν) −
ξ
n+1
(ξϕ)σ −1 ν − σ −1 (νϕ)ξ. Since ∇R
(σ −1 ν) = (ξσ −1 )ν + σ −1 ∇ξR
ξ
this gives Wxσ ξ = σ −1 Wx ξ − σ −1 (νϕ)ξ, which implies
n+1
ν, ∇R
ξ
n+1
ν = Wx ξ and ξσ −1 = σ −1 ξϕ,
Wxσ ξ = σ −1 Wx ξ − (νσ −1 )ξ = σ −1 (Wx − (νϕ))ξ.
(Here as before, νf = ∇ν f .) This gives the expression for the second fundamental form bσij = h∂xi , Wx ∂xj iσ =
σh∂xi , (Wx − (νϕ)∂xj i and therefore
bσij = e−ϕ (bij − (νϕ)gij ).
(A.21)
For the mean curvature Hσ = σ −2 gij bσij , we obtain
Hσ =
n
1
(H + 2 ∇ν σ) = eϕ (H − n∇ν ϕ).
σ
σ
(A.22)
We derive (A.22) differently using that the mean curvature arises in normal variations, η = f ν, of the
surface
volume functional V (ψ) (see (1.9)). In the conformal case, the analogue of (1.9) is dVσ (ψ)η =
R
√
√
√
√
√
H
ν
· η gσ dn u, where η = f νσ . We
use that gσ = σ n g, (d g)η
= (d g)(σ −1 f ν) = Hν · σ −1 f ν
σ
U
R
R
√
√
and νσ = σ −1 ν to compute dVσ (ψ)η = U (Hν + nσ −1 ∇σ) · η gσ dn u = U (H + nσ −1 ∇σ · ν)σ −1 f gσ dn u,
which implies Hσ = σ −1 (H + nσ −1 ∇ν σ).
Our next goal is to compute the normal hessian, HessN A, of the area functional A, for a hypersurface
S immersed in a manifold M = (Rn+1 σdx2 ), the Euclidean space Rn+1 , with the conformally flat metric
σdx2 , σ = e−2ϕ . Recall that the normal hessian, HessN A, of the area functional A, for a hypersurface
S immersed in a manifold M is given by the expression (4.26). Thus to to compute HessN A, we have to
compute the trace norm |WxM |2s , the Ricci curvature RicM (ν, ν) of M evaluated on the outward normal
vector field, ν,on S and the Laplace-Beltrami operator ∆S on S in the metric induced by the metric on M .
We begin with the trace norm |Wxσ |2σ = |WxM |2s . Since by the definition |Wxσ |2σ = σ −2 g ik g jm bσij bσkm , we
have |Wxσ |2σ = σ −1 g ik g jm (bij − ∇ν ϕgij )(bkm − ∇ν ϕgkm ), which gives
|Wxσ |2σ = e2ϕ (|Wx |2 − 2∇ν ϕH + n(∇ν ϕ)2 ).
(A.23)
Furthermore, to compute Ricσ = RicM , we use standard formulae, which give (see e.g. [46, 57])
Ricσij = (n − 1)(∇i ∇j ϕ − ∇i ϕ∇j ϕ) + (∆ϕ + (n − 1)|∇ϕ|2 )δij .
(A.24)
71
(Recall that ∇i , ∇, ∆ are the operators in the euclidean metric.)
Finally, we compute the change in the Laplace-Beltrami operator. By the equation (A.14) below, we
have
√
n
∂f
∂f
1
1
√
∆σS f = √ σ ∂ui ( g σ gσij j ) = n/2 √ ∂ui (σ 2 −1 gg ij j )
g
∂u
∂u
σ
g
n − 2 −2
n − 2 −2 j
∂f
= σ −1 ∆S +
σ (∂ui σ)g ij j = σ −1 ∆S f +
σ (∇ σ)∇j f
2
∂u
2
= e2ϕ (∆S − (n − 2)(∇j ϕ)∇j )f.
(A.25)
(A.26)
(A.27)
n
In the example of the n−sphere SR
of radius R in Rn+1 considered in Subsection ??, we computed
n
stand
stand
and Sn = S1n , respectively, which
, where gij and gij
bij = R−1 gij = Rgij
implies, in particular, that H = g ij bji = nR−1 . In this case, since ν = ω, (A.23) gives
|Wxσ |2σ = e2ϕ (nR−2 − 2nR−1 ∇ω ϕ + n(∇ω ϕ)2 ) = neϕ (R−1 − ∇ω ϕ)2 .
(A.28)
To compute Ricσ (ν σ , ν σ ), we use (A.24) and ν σ = σ −1/2 ν = σ −1/2 ω to obtain Ricσ (ν σ , ν σ ) = σ −1 Ricσ (ω, ω) =
e2ϕ [(n − 1)(∇i ∇j ϕ − ∇i ϕ∇j ϕ)ω i ω j + (∆ϕ + (n − 1)|∇ϕ|2 )].
Hence, due to (4.26), HessN A is given by
n
HessN Aσ (SR
) = −e2ϕ (R−2 ∆Sn − (n − 2)(∇j σ)∇j ) − ne2ϕ (R−1 − ∇ω ϕ)2
− e2ϕ [(n − 1) ∇i ∇j ϕω i ω j − (∇ϕ · ω)2 + |∇ϕ|2 + ∆ϕ],
(A.29)
n
HessN Aσ (SR
) = −e2ϕ R−2 (∆Sn + n) − (n − 2)(∇j σ)∇j − 2nR−1 ∇ω ϕ − (∇ω ϕ)2
+ (n − 1) ∇ω ∇ω ϕ + |∇ϕ|2 + ∆ϕ .
(A.31)
(A.30)
which gives
(A.32)
A.7
Riemann curvature tensor for hypersurfaces
A.8
Mean curvature of normal graphs over the round sphere Sn and cylinder
C n+1
Let Sˆ be a fixed convex n−dimensional hypersurface in Rn+1 . We consider hypersurfaces S given as normal
ˆ Here νˆ(ˆ
graphs θ(ˆ
x, t) = u(ˆ
x)ˆ
ν (ˆ
x), over a given hypersurface S.
x) is the outward unit normal vector to Sˆ at
ˆ
x
ˆ ∈ S. We would like to derive an expression for the mean curvature of S in terms of the graph function ρ.
To be more specific, we are interested in Sˆ being the round sphere Sn . Then a normal graph graph is
given by
θ : ω 7→ ρ(ω)ω.
(A.33)
The result is given in the next proposition.
Proposition 59. If a surface S, defined by an immersion θ : Sn → Rn+1 , is a graph (A.33) over the round
sphere Sn , with the graph function ρ : Sn → R+ , then the mean curvature, H, of S is given in terms of the
72
graph function u by the equation
H=p
−1/2
(n − ρ
−1
−3/2
∆ρ) + p
2
−1 ik j`
|∇ρ| + ρ g g ∇i ∇j ρ∇k ρ∇` ρ ,
(A.34)
where ∇ denotes covariant differentiation on Sn , with the components ∇i in some local coordinates and
p := ρ2 + |∇ρ|2 .
(A.35)
Proof. We do computations locally and therefore it is convenient to fix a point ω ∈ Sn and choose normal
coordinates at this point. Let (e1 , . . . , en ) denote an orthonormal basis of Tω Sn ⊂ Tω Rn+1 in this coordinates,
sphere
i.e. the metric on Sn at ω ∈ Sn is gij
= g sphere (ei , ej ) = δij and the Christphel symbols vanish, Γkij = 0
n
at ω ∈ S . By ∇i we denote covariant differentiation on Sn w.r.to this basis (frame). First, we compute the
metric g ij on St . To this end, we note that, for the map (A.33), one has
∇i θ = (∇i ρ)ω + uei , i = 1, . . . , n,
(A.36)
∂ψ
∂ψ
ij
where we used the relation W ∂u
i = g bjk ∂uk (see (A.7) of Appendix ??), where W is the Weigarten map,
sphere
which in our situation (i.e. bsphere
= gij
= δij and ν(ω) = ω) gives ∇i ω = ei . It follows that the
jk
Riemannian metric g induced on S has components
gij = h∇i θ, ∇j θi = ρ2 δij + ∇i ρ∇j ρ.
Recall that the symbol h·, ·i denotes the pointwise Euclidean inner product of Rn+1 . As can be easily checked
directly, the inverse matrix is
g ij = ρ−2 (δij − p−1 ∇i ρ∇j ρ).
(A.37)
It is easy to see that the following normalized vector field on S is orthogonal to all vectors (A.36) and
therefore is the outward unit normal to M is
ν = p−1/2 (ρω − ei ∇i ρ) = p−1/2 (ρω − ∇ρ).
(A.38)
Straightforward calculations show that
p1/2 ∇j ν = 2(∇j ρ)ω + ρej − ∇j ∇ρ + {· · · }ν,
where the terms in braces are easy to compute but irrelevant for what follows. One thus finds that the
matrix elements bij = h∇i θ, ∇j νi of the second fundamental form are
bij = p−1/2 (ρ2 δij + 2∇i ρ∇j ρ − ρ∇i ∇j ρ).
(A.39)
Since H = trg (h) = g ji bij , we use (A.37) and (A.39) and δji δij = n to compute:
H = p−1/2 [δji δij + 2ρ−2 δij ∇i ρ∇j ρ − ρ−2 δij ρ∇i ∇j ρ − p−1 ∇i ρ∇j ρδij − 2(ρ2 p)−1 ∇i ρ∇j ρ∇i ρ∇j ρ
+ (ρp)−1 ∇i ρ∇j ρ∇i ∇j ρ]
= p−1/2 [n + 2ρ−2 ∇i ρ∇i ρ − ρ−1 ∇i ∇i ρ − p−1 ∇i ρ∇i ρ − 2(ρ2 p)−1 |∇ρ|4 + (ρp)−1 ∇i ρ∇j ρ∇i ∇j ρ]
= p−1/2 [n + 2ρ−2 |∇ρ|2 − ρ−1 ∆ρ − p−1 |∇ρ|2 − 2(ρ2 p)−1 |∇ρ|4 + (ρp)−1 ∇i ρ∇j u∇i ∇j ρ]
= p−1/2 [n − ρ−1 ∆ρ + (ρp)−1 ∇i ρ∇j ρ∇i ∇j ρ + p−1 ρ−2 (2p|∇ρ|2 − ρ2 |∇ρ|2 − 2|∇ρ|4 )]
= p−1/2 [n − ρ−1 ∆ρ + ρ−1 p−1 ∇i ∇j ρ∇i ρ∇j ρ + p−1 |∇ρ|2 ].
which implies (A.34).
73
A different derivation is given below. (to be done)
B
B.1
Maximum principles
Elliptic maximum principle
Consider the equation −∆f + ∇v f + b = 0 on a compact manifold, M .
Theorem 60. Assume v and b are smooth and bounded and b > 0. Then f is a constant.
Proof. By compactness of M , f should reach a maximum on M . Let x∗ be a point of the maximum. Then
∇v f (x∗ ) = 0 and ∆f (x∗ ) ≤ 0. Therefore ∆f (x∗ ) = b(x∗ ) > 0, the contradiction.
B.2
Parabolic maximum principle
Theorem 61. Let u satisfy the differential inequality ∂t u ≥ ∆u + f (u) and let min u > 0 on a closed surface
S0 . Then u(t) satisfies
u − ϕ ≥ 0 for ∀t ∈ [0, T ),
(B.1)
where ϕ is x−independent and satisfies ∂t ϕ = f (ϕ) and ϕ|t=0 = min u|t=0 .
Proof. Let ϕ be as in the theorem. Then, since ϕ is independent of x and due to the differential inequality
for u,
∂t (u − ϕ) ≥ 4(H − ϕ) + f (u) − f (ϕ) and (u − ϕ)|t=0 ≥ 0,
(B.2)
which implies (B.1) Indeed, let u = ϕ at x = x
¯, t = t¯ and H > ϕ if either x 6= x
¯, t = t¯ or t < t¯. Then u|t=t¯
has a minimum at x = x
¯. So we obtain
Hess u(¯
x, t¯) > 0
4H(¯
x, t¯) = Tr Hess u(¯
x, t¯) > 0
¯
⇒ ∂t (u − ϕ) > f (u) − f (ϕ) = 0 at (¯
x, t)
⇒
(u − ϕ)(¯
x, t)
⇒
grows in t at
t¯.
This implies (B.1), as claimed.
C
C.1
Elements of spectral theory
A characterization of the essential spectrum of a Schr¨
odinger operator
In this appendix we present a result characterizing the essential spectrum of a Schrödinger operator in a
manner similar to the characterization of the discrete spectrum as a set of eigenvalues. We follow [?].
Theorem 62 (Schnol-Simon). Let H be a Schrödinger operator with a bounded potential. Then
σ(H) = closure {λ | (H − λ)ψ = 0 for ψ polynomially bounded }.
74
So we see that the essential spectrum also arises from solutions of the eigenvalue equation, but that
these solutions do not live in the space L2 (R3 ).
Proof. We prove only that the right hand side ⊂ σ(H), and refer the reader to [?] for a complete proof. Let
ψ be a polynomially bounded solution of (H − λ)ψ = 0. Let Cr be the box of side-length 2r centred at the
origin. Let jr be a smooth function with support contained in Cr+1 , with jr ≡ 1 on Cr , 0 ≤ jr ≤ 1, and
with supr,x,|α|≤2 |∂xα jr (x)| < ∞. Our candidate for a Weyl sequence is
wr :=
jr ψ
.
kjr ψk
Note that kwr k = 1. If ψ 6∈ L2 , we must have kjr ψk → ∞ as r → ∞. So for any R,
Z
Z
1
2
|ψ|2 → 0
|wr | ≤
kjr ψk2 |x|<R
|x|<R
as r → ∞. We show that
(H − λ)wr → 0.
Let F (r) =
R
Cr
2
|ψ| , which is monotonically increasing in r. We claim there is a subsequence {rn } such that
F (rn + 2)
→ 1.
F (rn − 1)
If not, then there is a > 1 and r0 > 0 such that
F (r + 3) ≥ aF (r)
for all r ≥ r0 . Thus F (r0 + 3k) ≥ ak F (r0 ) and so F (r) ≥ (const)br with b = a1/3 > 1. But the assumption
that ψ is polynomially bounded implies that F (r) ≤ (const)rN for some N , a contradiction. Now,
(H − λ)jr ψ = jr (H − λ)ψ + [−∆, jr ]ψ.
Since (H − λ)ψ = 0 and [∆, jr ] = (∆jr ) + 2∇jr · ∇, we have
(H − λ)jr ψ = (−∆jr )ψ − 2∇jr · ∇ψ.
Since |∂ α jr | is uniformly bounded,
Z
k(H − λ)jr ψk ≤ (const)
(|ψ|2 + |∇ψ|2 ) ≤ (const)
Cr+1 \Cr
So
k(H − λ)wr k ≤ C
Z
Cr+1 Cr
F (r + 2)
F (r + 2) − F (r − 1)
≤ C(
− 1)
F (r)
F (r − 1)
and so k(H − λ)wrn k → 0. Thus {wrn } is a Weyl sequence for H and λ. |ψ|2 .
C.2
75
Perron-Frobenius Theory
Consider a bounded operator T on the Hilbert space X = L2 (Ω).
Definition 6. An operator T is called positivity preserving/improving if and only if u ≥ 0, u 6= 0 =⇒
T u ≥ 0/T u > 0.
Note if T is positivity preserving, then T maps real functions into real functions.
Theorem 63. Let T be a bounded positive and positivity improving operator and let λ be an eigenvalue of
T with an eigenvector ϕ. Then
a) λ = kT k ⇒ λ is simple and ϕ > 0 (modulo a constant factor).
b) ϕ > 0 and kT k is an eigenvalue of T ⇒ λ is simple and λ = kT k.
Proof. a) Let λ = kT k, T ψ = λψ and ψ be real. Then |ψ| ± ψ ≥ 0 and therefore T (|ψ| ± ψ) > 0. The latter
inequality implies that |T ψ| ≤ T |ψ| and therefore
h|ψ|, T |ψ|i ≥ h|ψ|, |T ψ|i ≥ hψ, T ψi = λkψk2 .
Since λ = kT k = supkψk=1 hψ, T ψi, we conclude using variational calculus (see e.g. [?] or [65]) that
T |ψ| = λ|ψ|
(C.1)
i.e., |ψ| is an eigenfunction of T with the eigenvalue λ. Indeed, since λ = kT k = supkψk=1 hψ, T ψi, |ψ| is the
maximizer for this problem. Hence |ψ| satisfies the Euler-Lagrange equation T |ψ| = µ|ψ| for some µ. This
implies that µkψk2 = h|ψ|, T |ψ|i = λkψk2 and hence µ = λ. Equation (C.1) and the positivity improving
property of T imply that |ψ| > 0.
Now either ψ = ±|ψ| or |ψ| + ψ and |ψ| − ψ are nonzero. In the latter case they are eigenfunctions of T
corresponding to the eigenvalue λ : T (|ψ| ± ψ) = λ(|ψ| ± ψ). By the positivity improving property of T this
implies that |ψ| ± ψ > 0 which is impossible. Thus ψ = ±|ψ|.
If ψ1 and ψ2 are two real eigenfunctions of T with the eigenvalue λ then so is aψ1 + bψ2 for any a, b ∈ R.
By the above, either aψ1 + bψ2 > 0 or aψ1 + bψ2 < 0 ∀a, b ∈ R \ {0}, which is impossible. Thus T has a
single real eigenfunction associated with λ.
Let now ψ be a complex eigenfunction of T with the eigenvalue λ and let ψ = ψ1 + ıψ1 where ψ1 and
ψ2 are real. Then the equation T ψ = λψ becomes
T ψ1 + iT ψ2 = λψ1 + iλψ2 .
Since T ψ2 and T ψ2 and λ are real (see above) we conclude that T ψi = λψ2 , i = 1, 2, and therefore by the
above ψ2 = cψ1 for some constant c. Hence ψ = (1 + ıc)ψ1 is positive and unique modulo a constant complex
factor.
b) By a) and eigenfunction, ψ, corresponding to ν := kT k can be chosen to be positive, ψ > 0. But then
76
λhψ, ϕi = hψ, T ϕi = hT ψ, ϕi = νhψ, ϕi
and therefore λ = ν and ψ = cϕ.
*Question: Can the condition that kT k is an eigenvalue of T (see b) be removed?*
Now we consider the Schr¨
odinger operator H = −∆ + V (x) with a real, bounded potential V (x). The
above result allows us to obtain the following important
Theorem 64. Let H = −∆ + V (x) have an eigenvalue E0 with an eigenfunction ϕ0 (x) and let inf σ(H) be
an eigenvalue. Then
ϕ0 > 0 ⇒ E0 = inf{λ|λ ∈ σ(H)} and E0 is non-degenerate
and, conversely,
E0 = inf{λ|λ ∈ σ(H)} ⇒ E0 is non-degenerate and ϕ0 > 0
(modulo multiplication by a constant factor).
Proof. To simplify the exposition we assume V (x) ≤ 0 and let W (x) = −V (x) ≥ 0. For µ > sup W we have
(−∆ − W + µ)−1 = (−∆ + µ)−1
∞
X
[W (−∆ + µ)−1 ]n
(C.2)
n=0
where the series converges in norm as
kW (−∆ + µ)−1 k ≤ kW kk(−∆ + µ)−1 k ≤ kW kL∞ µ−1 < 1
by our assumption that µ > sup W = kW kL∞ . To be explicit we assume that d = 3. Then the operator
(−∆ + µ)−1 has the integral kernel
√
e− µ|x−y|
>0
4π|x − y|
while the operator W (−∆ + µ)−1 has the integral kernel
√
e− µ|x−y|
W (x)
≥ 0.
4π|x − y|
Consequently, the operator
(−∆ + µ)
−1
1
f (x) =
4π
Z
√
e− µ|x−y|
f (y) dy
|x − y|
is positivity improving (f ≥ 0, f 6= 0 ⇒ (−∆ + µ)−1 f > 0) while the operator
77
W (−∆ + µ)−1 f (x) =
1
4π
√
Z
W (x)
e− µ|x−y|
f (y) dy
|x − y|
is positivity preserving (f ≥ 0 ⇒ W (−∆ + µ)−1 f ≥ 0). The latter fact implies that the operators [W (−∆ +
µ)−1 ]n , n ≥ 1, are positivity preserving (prove this!) and consequently the operator
(−∆ + µ)−1 +
∞
X
[W (−∆ + µ)−1 ]n
n=1
is positivity improving (prove this!).
Thus we have shown that the operator (H + µ)−1 is positivity improving. Series (C.2) shows also that
(H + µ)−1 is bounded. Since
hu, (H + µ)ui ≥ (− sup W + µ)kuk2 > 0,
we conclude that the operator (H + µ)−1 is positive (as an inverse of a positive operator). Finally, k(H +
µ)−1 k = sup σ((H + µ)−1 ) = (inf σ(H) + µ)−1 is an eigenvalue by the condition of the theorem. Hence the
previous theorem applies to it. Since Hϕ0 = E0 ϕ0 ⇔ (H + µ)−1 ϕ0 = (E0 + µ)−1 ϕ0 , the theorem under
verification follows. *This paragraph needs details!*.
D
Proof of the Feynmann-Kac Formula
In this appendix we present, for the reader’s convenience, a proof of the Feynman-Kac formula ( 6.26)-( 6.27)
and the estimate (6.29).
2
Let L0 := −∂y2 + α4 y 2 − α2 and L := L0 + V where V is a multiplication operator by a function V (y, τ ),
which is bounded and Lipschitz continuous in τ . Let U (τ, σ) and U0 (τ, σ) be the propagators generated
by the operators −L and −L0 , respectively. The integral kernels of these operators will be denoted by
U (τ, σ)(x, y) and U0 (τ, σ)(x, y).
Theorem 65. The integral kernel of U (τ, σ) can be represented as
Z R
τ
U (τ, σ)(x, y) = U0 (τ, σ)(x, y) e σ V (ω0 (s)+ω(s),s)ds dµ(ω)
(D.1)
where dµ(ω) is a probability measure (more precisely, a conditional harmonic oscillator, or Ornstein-Uhlenbeck,
probability measure) on the continuous paths ω : [σ, τ ] → R with ω(σ) = ω(τ ) = 0, and the pathe ω0 (·) is
given by
e2ασ − e2αs
e2ατ − e2αs
ω0 (s) = eα(τ −s) 2ασ
x + eα(σ−s) 2ατ
y.
(D.2)
2ατ
e
−e
e
− e2ασ
Remark 7. dµ(ω) is the Gaussian measure with mean zero and covariance (−∂s2 + α2 )−1 , normalized to 1.
The path ω0 (s) solves the boundary value problem
(−∂s2 + α2 )ω0 = 0 with ω(σ) = y and ω(τ ) = x.
(D.3)
78
Proof of Theorem 65. We begin with the following extension of the Ornstein-Uhlenbeck process-based FeynmanKac formula to time-dependent potentials:
Z R
τ
U (τ, σ)(x, y) = U0 (τ, σ)(x, y) e σ V (ω(s),s) ds dµxy (ω).
(D.4)
where dµxy (w) is the conditional Ornstein-Uhlenbeck probability measure, which is the normalized Gaussian
measure dµxy (ω) with mean ω0 (s) and covariance (−∂s2 + α2 )−1 , on continuous paths ω : [σ, τ ] → R with
ω(σ) = y and ω(τ ) = x (see e.g. [31, ?, ?]). This formula can be proven in the same way as the one for time
independent potentials (see [31], Equation (3.2.8)), i.e. by using the Kato-Trotter formula and evaluation
of Gaussian measures on cylindrical sets. Since its proof contains a slight technical wrinkle, for the reader’s
convenience we present it below.
Now changing the variable of integration in (D.4) as ω = ω0 + ω
˜ , where ω
˜ (s) is a continuous
path with
R
boundary
conditions
ω
˜
(σ)
=
ω
˜
(τ
)
=
0,
using
the
translational
change
of
variables
formula
f
(ω)
dµ
xy (ω) =
R
f (ω0 + ω
˜ ) dµ(˜
ω ), which can be proven by taking f (ω) = eihω,ζi and using (D.3) (see [31], Equation (9.1.27))
and omitting the tilde over ω we arrive at (D.1).
There are at least three standard ways to prove (D.4): by using the Kato-Trotter formula, by expanding
both sides of the equation in V and comparing the resulting series term by term and by using Ito’s calculus
(see [?, ?, 64, 31]). The first two proofs are elementary but involve tedious estimates while the third proof is
based on a fair amount of stochastic calculus. For the reader’s convenience, we present the first elementary
proof of (D.4).
Lemma 66. Equation (D.4) holds.
Proof. In order to simplify our notation, in the proof that follows we assume, without losing generality,
that σ = 0. We divide the proof into two parts. First we prove that for any fixed ξ ∈ C0∞ the following
Kato-Trotter type formula holds
Y
U (τ, 0)ξ = lim
n→∞
0≤k≤n−1
k+1 k
U0 (
τ, τ )e
n
n
R
(k+1)τ
n
kτ
n
V (y,s)ds
ξ
(D.5)
in the L2 space. We start with the formula
Y
U (τ, 0) −
0≤k≤n−1
=
Y
0≤k≤n−1
=
X
k+1 k
U0 (
τ, τ )e
n
n
k+1 k
U(
τ, τ ) −
n
n
Y
0≤j≤n j≤k≤n−1
U0 (
R
(k+1)τ
n
kτ
n
k+1 k
U0 (
τ, τ )e
n
n
Y
0≤k≤n−1
R (k+1)τ
n
k+1 k
τ, τ )e
n
n
V (y,s)ds
kτ
n
V (y,s)ds
R
(k+1)τ
n
kτ
n
j
Aj U ( τ, 0)
n
with the operator
j+1 j
Aj := U0 (
τ, τ )e
n
n
R
(j+1)τ
n
jτ
n
V (y,s)ds
− U(
j+1 j
τ, τ ).
n
n
V (y,s)ds
79
We observe that kU0 (τ, σ)kL2 →L2 ≤ 1, and moreover by the boundness of V, the operator U (τ, σ) is
uniformly bounded in τ and σ in any compact set. Consequently
Y
k[U (τ, 0) −
0≤k≤n−1
Y
≤ max nk
j
j≤k≤n−1
k+1 k
U0 (
τ, τ )e
n
n
k+1 k
U0 (
τ, τ )e
n
n
R
R
(k+1)τ
n
kτ
n
(k+1)τ
n
kτ
n
V (y,s)ds
V (y,s)ds
]ξk2
j
Aj U ( τ, 0)ξk2
n
(D.6)
j
. max nkAj U ( τ, 0)ξk2 .
j
n
We show below that
j
max nkAj U ( τ, 0)ξk2 → 0,
j
n
as n → ∞. Equations (D.7), ( D.6), ( D.13) and imply ( D.5). This completes the first step.
(D.7)
In the second step we compute the integral kernel, Gn (x, y), of the operator
Gn :=
Y
0≤k≤n−1
k+1 k
U0 (
τ, τ )e
n
n
R
(k+1)τ
n
kτ
n
V (·,s)ds
in ( D.5). By the definition, Gn (x, y) can be written as
Z
Gn (x, y) =
Z
···
R
Y
U (xk+1 , xk )e
τ
n
(k+1)τ
n
kτ
n
V (xk ,s)ds
dx1 · · · dxn−1
(D.8)
0≤k≤n−1
with xn := x, x0 := y and Uτ (x, y) ≡ U0 (0, τ )(x, y) is the integral kernel of the operator U0 (τ, 0) = e−L0 τ .
We rewrite (D.8) as
Pn−1 R
Z
Gn (x, y) = Uτ (x, y)
where
k=0
e
Q
0≤k≤n−1
dµn (x1 , . . . , xn ) :=
Since Gn (x, y)|V =0 = Uτ (x, y) we have that
interval in R. Define a cylindrical set
(k+1)τ
n
kτ
n
V (xk ,s) ds
U nτ (xk+1 , xk )
Uτ (x, y)
R
dµn (x1 , . . . , xn ),
(D.9)
dx1 . . . dxk−1 .
dµn (x1 , . . . , xn ) = 1. Let ∆ := ∆1 × . . . × ∆n , where ∆j is an
n
P∆
:= {ω : [0, τ ] → R | ω(0) = y, ω(τ ) = x, ω(kτ /n) ∈ ∆k , 1 ≤ k ≤ n − 1}.
R
n
By the definition of the measure dµxy (ω), we have µxy (P∆
) = ∆ dµn (x1 , . . . , xn ). Thus, we can rewrite
(D.9) as
Z Pn−1 R (k+1)τ
n
V (ω( kτ
n ),s) ds
kτ
k=0
n
dµxy (ω),
(D.10)
Gn (x, y) = Uτ (x, y) e
By the dominated convergence theorem the integral on the right hand side of (D.10) converges in the sense
of distributions as n → ∞ to the integral on the right hand side of (D.4). Since the left hand side of (D.10)
converges to the left hand side of (D.4), also in the sense of distributions (which follows from the fact that
Gn converges in the operator norm on L2 to U (τ, σ)), (D.4) follows.
80
Now we prove (D.7). We have
j
k 1
j 1
j
max nkAj U ( τ, 0)ξk2 . n max kAj + K( τ, τ )kL2 →L2 + max nkK( τ, τ )U ( τ, 0)ξk2 ,
j
j
j
n
n n
n n
n
where the operator K(σ, δ) is defined as
Z δ
Z
K(σ, δ) :=
U0 (σ + δ, σ + s)V (σ + s, ·)U0 (σ + s, σ)ds − U0 (σ + δ, σ)
0
(D.11)
δ
V (σ + s, ·)ds
(D.12)
0
Now we claim that
k 1
1
kAj + K( τ, τ )kL2 →L2 . 2 ,
n n
n
(D.13)
1
sup k K(σ, δ)U (σ, 0)ξk2 → 0.
0≤σ≤τ δ
(D.14)
and
We begin with proving the first estimate. By Duhamel’s principle we have
j+1 j
j+1 j
U(
τ, τ ) = U0 (
τ, τ ) +
n
n
n
n
Z
j+1
n τ
j
nτ
U0 (
j
j+1
τ, s)V (y, s)U (s, τ )ds.
n
n
Iterating this equation on U (s, nk τ ) and using the fact that U (s, t) is uniformly bounded if s, t is on a compact
set, we obtain
Z n1 τ
j+1 j
j
1
j+1 j
j+1
τ, τ ) − U0 (
τ, τ ) −
τ, s)V (y, s)U0 (s, τ )dskL2 →L2 . 2 .
kU (
U0 (
n
n
n
n
n
n
n
0
R
On the other hand we expand e
j+1 j
kU0 (
τ, τ )e
n
n
R
(j+1)τ
n
jτ
n
(j+1)τ
n
jτ
n
V (y,s)ds
V (y,s)ds
and use the fact that V is bounded to get
j+1 j
j+1 j
− U0 (
τ, τ ) − U0 (
τ, τ )
n
n
n
n
Z
(j+1)τ
n
jτ
n
V (y, s)dskL2 →L2 .
1
.
n2
By the definition of K and Aj we complete the proof of ( D.13). A proof of (D.14) is given in the lemma
that follows.
Lemma 67. For any σ ∈ [0, τ ] and ξ ∈ C0∞ we have, as δ → 0+ ,
1
sup k K(σ, δ)U (σ, 0)ξk2 → 0.
0≤σ≤τ δ
(D.15)
Proof. If the potential term, V , is independent of τ , then the proof is standard (see, e.g. [64]). We use
the property that the function V is Lipschitz continuous in time τ to prove (D.15). The operator K can be
further decomposed as
K(σ, δ) = K1 (σ, δ) + K2 (σ, δ)
with
Z
K1 (σ, δ) :=
δ
U0 (σ + δ, σ + s)V (σ, ·)U0 (σ + s, σ)ds − δU0 (σ + δ, σ)V (σ, ·)
0
81
and
Z
K2 (σ, δ) :=
δ
Z
δ
U0 (σ + δ, σ + s)[V (σ + s, ·) − V (σ, ·)]U0 (σ + s, σ)ds − U0 (σ + δ, σ)
0
[V (σ + s, ·) − V (σ, ·)]ds.
0
Since U0 (τ, σ) are uniformly L2 -bounded and V is bounded, we have U (τ, σ) is uniformly L2 -bounded.
This together with the fact that the function V (τ, y) is Lipschitz continuous in τ implies that
Z δ
kK2 (σ, δ)kL2 →L2 . 2
sds = δ 2 .
0
We rewrite K1 (σ, δ) as
Z δ
U0 (σ + δ, σ + s){V (σ, ·)[U0 (σ + s, σ) − 1] − [U0 (σ + s, σ) − 1]V (σ, ·)}ds.
K1 (σ, δ) =
0
Let ξ(σ) = U (σ, 0)ξ. We claim that for a fixed σ ∈ [0, τ ],
kK1 (σ, δ)ξ(σ)k2 = o(δ).
(D.16)
Indeed, the fact ξ0 ∈ C0∞ implies that L0 ξ(σ), L0 V (σ)ξ(σ) ∈ L2 . Consequently (see [63])
lim+
s→0
(U0 (σ + s, σ) − 1)g
→ L0 g,
s
for g = ξ(σ) or V (σ, y)ξ(σ) which implies our claim. Since the set of functions {ξ(σ)|σ ∈ [0, τ ]} ⊂ L0 L2 is
compact and k 1δ K1 (σ, δ)kL2 →L2 is uniformly bounded, we have (D.16) as δ → 0 uniformly in σ ∈ [0, τ ].
Collecting the estimates on the operators Ki , i = 1, 2, we arrive at ( D.15).
Note that on the level of finite dimensional approximations the change of variables formula can be derived
as follows. It is tedious, but not hard, to prove that
Y
Un (xk+1 , xk ) = e
−α
0≤k≤n−1
(x−e−ατ y)2
2(1−e−2ατ )
Y
Un (yk+1 , yk )
0≤k≤n−1
with yk := xk − ω0 ( nk τ ). By the definition of ω0 (s) and the relations x0 = y and xn = x we have
Gn (x, y) = Uτ (x, y)G(1)
n (x, y)
(D.17)
where
G(1)
n (x, y)
1
√
:=
4π α(1 − e−2ατ )
Z
Z
···
Y
R
Un (yk+1 , yk )e
(k+1)τ
n
kτ
n
V (yk +ω0 ( kτ
n ),s)ds
dy1 · · · dyk−1 . (D.18)
0≤k≤n−1
Since lim Gn ξ exists by ( D.15), we have lim G(1)
n ξ (in the weak limit) exists also. As shown in [31],
n→∞ Z
n→∞
Rτ
lim G(1)
e 0 V (ω0 (s)+ω(s),s)ds dµ(ω) with dµ being the (conditional) Ornstein-Uhlenbeck measure on
n =
n→∞
the set of path from 0 to 0. This completes the derivation of the change of variables formula.
Remark 8. In fact, Equations (D.5), (D.17) and (D.18) suffice to prove the estimate in Corollary 42.
82
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Lectures on Mean Curvature Flow (MAT 1063 HS) I. M. Sigal

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