Branching geodesics in metric spaces with Ricci curvature

Transcription

Optimal mass transportation
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
Branching geodesics in metric spaces with Ricci
curvature lower bounds
Tapio Rajala
University of Jyväskylä
tapio.m.rajala@jyu.fi
Interactions Between Analysis and Geometry:
Analysis on Metric Spaces
IPAM, UCLA, Mar 18th, 2013
Tapio Rajala
Branching geodesics in spaces with Ricci curvature lower bounds
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
in geodesic metric spaces
The space (P(X ), W2)
For µ, ν ∈ P(X ) define
Z
W2 (µ, ν) = inf
X ×X
X
1/2
(p1 )# σ = µ
d(x, y ) dσ(x, y ) .
(p2 )# σ = ν
2
ν
σ
µ
Tapio Rajala
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
Z
W2 (µ, ν) = inf
X ×X
X
1/2
(p1 )# σ = µ
d(x, y ) dσ(x, y ) .
(p2 )# σ = ν
2
σ
ν
µ
Tapio Rajala
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
Z
W2 (µ, ν) = inf
X ×X
X
1/2
(p1 )# σ = µ
d(x, y ) dσ(x, y ) .
(p2 )# σ = ν
2
ν
µ
Tapio Rajala
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
Question (Existence of optimal maps)
When is the/an optimal plan induced by a map? (σ = (id , T )# µ
for some map T : X → X )
The usual steps in the proof are (given two measures
µ, ν ∈ P(X )):
1
Prove that any optimal plan from µ to ν is induced by a map.
2
Linear combinations of optimal plans are optimal.
3
The combination of two different optimal maps is not a map.
Hence there is only one optimal map.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
Existence of optimal maps
The existence of optimal maps for the quadratic cost has been
obtained for example by
Brenier (1991) in Rn with µ ≪ Ln . (Earlier steps by Brenier
1987 and Knott & Smith 1984)
McCann (1995) in Rn with µ(E ) = 0 for n − 1-rectifiable E .
Gangbo & McCann (1996) in Rn with µ(E ) = 0 for all
c − c-hypersurfaces E .
McCann (2001) for Riemannian manifolds M with µ ≪ vol.
Gigli (2011) for Riemannian manifolds M there exists an
optimal map from µ to every µ if and only if µ(E ) = 0 for all
c − c-hypersurfaces E .
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
Existence of optimal maps
and . . .
Bertrand (2008) for finite dimensional Alexandrov spaces with
µ ≪ Hd .
Gigli (2012) for non-braching CD(K , N)-spaces, N < ∞, with
µ ≪ m. For non-branching CD(K , ∞)-spaces with
µ, ν ∈ D(Entm ).
Rajala & Sturm (2012) for RCD(K , ∞)-spaces with µ, ν ≪ m.
Cavalletti & Huesmann (2013) for non-branching
MCP(K , N)-spaces with µ ≪ m.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
The space (P(X ), W2) for (X , d ) geodesic
(All the geodesics in this talk are constant speed geodesics
parametrized by [0, 1].)
If (X , d) is geodesic, then (P(X ), W2 ) is geodesic.
A geodesic (µt ) ⊂ Geo(P(X )) can be realized as a probability
measure π ∈ P(Geo(X )) in the sense that for all t ∈ [0, 1] we
have (et )# π = (µt ).
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
From Geo(P(X )) to P(Geo(X ))
Optimal plan between µ0 and µ1 as a geodesic in P(X ).
µ0
µ1
µ1
2
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
From Geo(P(X )) to P(Geo(X ))
Optimal plan between µ0 and µ1 as a measure in P(Geo(X )).
µ1
µ0
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
in metric spaces
Optimal plans σ vs. Geo(P(X )) vs. P(Geo(X ))
Denote by OptGeo(µ0 , µ1 ) ⊂ P(Geo(X )) the set of all π for
which (et )# π is a geodesic connecting µ0 and µ1 . The space
OptGeo(µ0 , µ1 ) has the most information on transports between
µ0 and µ1 : Usually neither of the maps
OptGeo(µ0 , µ1 ) → Geo(P(X )) : π 7→ (t 7→ (et )# π),
OptGeo(µ0 , µ1 ) → P(X × X )) : π 7→ (e0 , e1 )# π
is injective.
Moreover, a geodesic (µt )1t=0 does not (in general) define an
optimal plan σ, nor does σ define (µt )1t=0 .
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
branching vs. non-branching
local Poincaré inequalities
Optimal transport on Riemannian manifolds
zero curvature
Curvature changes the size of the
support of the transported mass.
positive curvature
Tapio Rajala
negative curvature
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Theorem (Otto & Villani 2000, Cordero-Erausquin, McCann &
Schmuckenschläger 2001, von Renesse & Sturm 2005)
For any smooth connected Riemannian manifold M and any
K ∈ R the following properties are equivalent:
1
Ric(M) ≥ K in the sense that Ricx (v , v ) ≥ K |v |2 for all
x ∈ M and v ∈ Tx M.
Entvol is K -convex on P2 (M).
R
Here Entvol (ρvol) = M ρ log ρ dvol and K -convexity of Entvol on
P2 (M) means that along every geodesic (µt )1t=0 ⊂ P2 (M) we have
2
Entvol (µs ) ≤ (1 − s)Entvol (µ0 ) + sEntvol (µ1 )
K
− s(1 − s)W22 (µ0 , µ1 )
2
for all s ∈ [0, 1].
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Ricci-curvature bounds in metric spaces
Definition (Sturm (2006))
(X , d, m) is a CD(K , ∞)-space (K ∈ R) if between any two
absolutely continuous measures there exists a geodesic
(µt ) ∈ Geo(P(X
R )) along which the entropy
Entm (ρm) = ρ log ρ dm is K -convex:
Entm (µs ) ≤ (1 − s)Entm (µ0 ) + sEntm (µ1 )
K
− s(1 − s)W22 (µ0 , µ1 )
2
for all s ∈ [0, 1].
There is also a slightly stronger definition by Lott and Villani.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Ricci limit spaces
Riemannian manifolds with Ric ≥ K
CD(K , ∞) of Lott-Villani
CD(K , ∞) of Sturm
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Branching geodesics
One problematic feature of the definition is that CD(K , ∞)-spaces
include also spaces with branching geodesics; like (R2 , || · ||∞ , L2 ).
We say that two geodesics γ 1 6= γ 2 , branch if there exists
t0 ∈ (0, 1) so that γt1 = γt2 for all t ∈ [0, t0 ].
γ01 = γ02
γ1
γt10 = γt20
γ2
A space where there are no branching geodesics is called
non-branching.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Ricci limit spaces
CD(K , ∞) + non-branching
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Localization without branching
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Localization with branching
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Localization in time with branching
Entm
µt
µ0
µs
Tapio Rajala
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µt1
µs
Tapio Rajala
µt2
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µt1
µs
Tapio Rajala
µt2
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µ1
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µ1
2
Tapio Rajala
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µ1
4
Tapio Rajala
µ1
2
µ3
4
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µ1
4
Tapio Rajala
µ1
2
µ3
4
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
µt
µ0
µ1
4
Tapio Rajala
µ1
2
µ3
4
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Not only does the new geodesic satisfy the K -convexity inequality
between any three times 0 < t1 < s < t2 < 1, but also the
measures along the geodesic have bounded densities (under some
assumptions on the initial and final measures). This is true for
instance if we start with two measures µ0 and µ1 having bounded
densities and bounded supports.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Iterating minimization fails for CD(K , N)
R
In CD(K , N)-spaces, minimizing the (Rényi) entropy X ρ1−1/N dm
does not always produce a geodesic along which the inequalities
required by CD(K , N)-condition hold.
Entm
µt
µ0
µ1
4
Tapio Rajala
µ1
2
µ1
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Iterating minimization fails for CD(K , N)
R
In CD(K , N)-spaces, minimizing the (Rényi) entropy X ρ1−1/N dm
does not always produce a geodesic along which the inequalities
required by CD(K , N)-condition hold.
However, if one of the measures is singular with respect to m, we
get the correct CD(K , N)-bounds when taking intermediate points
towards this measure.
Combining this observation with the density bounds gives
Theorem
CD(K , N) ⇒ MCP(K , N) (by Ohta).
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Definition
A space (X , d, m) is said to satisfy the measure contraction
property MCP(K , N) (inpthe sense of Ohta) if for every x ∈ X and
A ⊂ X (and A ⊂ B(x, π (N − 1)/K ) if K > 0) with
0 < m(A) < ∞ there exists
1
π ∈ GeoOpt δx ,
m|A
m(A)
so that
dm ≥ (et )# t N βt (γ0 , γ1 )m(A)dπ(γ) .
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
local Poincaré inequality in non-branching MCP(K , N)
Theorem (Lott & Villani, von Renesse, Sturm, Hinde & Petersen,
Cheeger & Colding)
Suppose that (X , d, m) is a nonbranching MCP(K , N)-space with
K ∈ R. Then the weak local Poincaré inequality
Z
Z
g dm
|u − huiB(x,r ) | dm ≤ C (N, K , r )r −
−
B(x,2r )
B(x,r )
holds for any measurable function u defined on X , any upper
gradient g of u and for each point x ∈ X and radius r > 0.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
local Poincaré inequality in CD(K , ∞)
Theorem
Suppose that (X , d, m) is a CD(K , ∞)-space (in the sense of
Sturm) with K ∈ R. Then the weak local Poincaré inequality
Z
Z
K −r 2
g dm
|u − huiB(x,r ) | dm ≤ 4re
B(x,2r )
B(x,r )
holds for any measurable function u defined on X , any upper
gradient g of u and for each point x ∈ X and radius r > 0.
Observe that we do not have average integrals in this theorem.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Proof of the local Poincaré inequality
Let us prove the CD(0, ∞) case for simplicity. We have to show
that
Z
Z
g dm.
|u − huiB(x,r ) | dm ≤ 4r
B(x,2r )
B(x,r )
Abbreviate B = B(x, r ) and denote
m(B)
M = inf a ∈ R : m({u > a}) ≤
.
2
Split the ball B into two Borel sets B + and B − so that
B = B + ∪ B − , B + ∩ B − = ∅, m(B + ) = m(B − ) and
u(x) ≤ M ≤ u(y )
Tapio Rajala
for all x ∈ B − , y ∈ B + .
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Let (µt )1t=0 be a geodesic between m(B1 + ) m|B + and m(B1 − ) m|B −
along which we have the density bound (writing µt = ρt m)
ρt (y ) ≤
2
m(B)
for all t ∈ [0, 1] at m-almost every y ∈ X . Let π be a
corresponding measure on the set of geodesics.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
From u(z) ≤ M ≤ u(y ) for all (z, y ) ∈ B − × B + we get
|u(γ(0)) − u(γ(1))| = |u(γ(0)) − M| + |M − u(γ(1))|
for π-almost every γ ∈ Geo(X ). Therefore
Z
|u(γ(0)) − u(γ(1))| dπ(γ)
Geo(X )
Z
Z
|M − u(γ(1))| dπ(γ)
|u(γ(0)) − M| dπ(γ) +
=
Geo(X )
Geo(X )
Z
Z
2
2
|u(z) − M| dm(z) +
|M − u(z)| dm(z)
=
m(B) B +
m(B) B −
Z
2
=
|u(z) − M| dm(z).
m(B) B
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
ZZ
1
|u − huiB(x,r ) | dm ≤
|u(z) − u(y )| dm(z) dm(y )
m(B) B×B
B(x,r )
ZZ
1
(|u(z) − M| + |M − u(y )|) dm(z) dm(y )
≤
m(B) B×B
Z
Z
|u(γ(0)) − u(γ(1))| dπ(γ)
|u(z) − M| dm(z) = m(B)
=2
Z
Geo(X )
B
≤ 2r m(B)
= 2r m(B)
Z 1Z
≤ 4r
0
Z
Geo(X )
Z 1Z
0
Z
1
g (γ(t)) dt dπ(γ)
0
g (z)ρt (z) dm(z) dt
Z
g (z) dm(z) dt = 4r
X
B(x,2r )
Tapio Rajala
g dm.
B(x,2r )
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
essential non-branching
Ricci limit spaces
CD(K , ∞) + non-branching
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Question
Does there exist a stable definition of Ricci curvature lower bounds
that excludes branching?
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
RCD(K , ∞)
Definition (Ambrosio, Gigli & Savaré, 2011 (preprint))
A metric measure space (X , d, m) has Riemannian Ricci curvature
bounded below by K ∈ R, or RCD(K , ∞) for short, if one of the
following equivalent conditions hold:
1
2
3
(X , d, m) isRa CD(K , ∞) space and the Cheeger-energy
Ch(f ) = 12 |Df |2w is a quadratic form on L2 (X , m).
(X , d, m) is a CD(K , ∞) space and the W2 gradient flow of
Entm is additive on P2 (X ).
Any µ ∈ P2 (X ) is the starting point of an EVIK gradient flow
of Entm .
Theorem (Daneri & Savaré, 2008)
EVIK implies strong displacement K -convexity.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
RCD(K , ∞) ⇒ essential non-branching
Theorem (R. & Sturm, 2012 (preprint))
Strong CD(K , ∞)-spaces are essentially non-branching. In
particular RCD(K , ∞)-spaces are essentially non-branching.
Corollary (of essential non-branching and Gigli’s result)
There exist optimal transport maps in strong CD(K , ∞)-spaces
between µ0 , µ1 ≪ m.
Definition
A space (X , d, m) is called essentially non-branching if for every
µ0 , µ1 ∈ P2 (X ) that are absolutely continuous with respect to m
we have that any π ∈ OptGeo(µ0 , µ1 ) is concentrated on a set of
non-branching geodesics. (Meaning that π(Γ) = 1 for some
Γ ⊂ Geo(X ) so that there are no two branching geodesics in Γ.)
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Ricci limit spaces
RCD(K , ∞)
CD(K , ∞) + essential non-branching
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
“Proof” Suppose that the claim is not true so that there exists a
measure π that is not concentrated on non-branching geodesics.
By restricting the measure π we may assume that there are
0 < t1 < t2 < 1 with |t1 − t2 | small and two sets of geodesics
Γ1 , Γ2 so that
(et )# π|Γ1 = (et )# π|Γ2
for all t ∈ [0, t1 ]
(et )# π|Γ1 ⊥ (et )# π|Γ2
for all t ∈ [t2 , 1]
and
for all t ∈ [t2 , 1].
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
0
t1
definition
t2
1
Γ1
Γ2
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
0
t1
t2
1
Γ1
Γ2
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
0
t1
t2
1
Γ1
Γ2
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Entm
log 2
0
t1
t2
1
Γ1
Γ2
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
ess. nb.
Ricci-limits
RCD(K , N)
non-branching
CD(K , N)
CD(K , N)
∃ maps
Local Poincaré
MCP(K , N)
MCP(K , N)
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Why is ||ρ1/2||∞ ≤ max{||ρ0 ||∞ , ||ρ1 ||∞ } =: M?.
ρ0
ρ1
ρ1
2
M
M
M
A
X
X
Tapio Rajala
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Consider the curve Γ ∈ P(Geo(X )) between the marginals
corresponding to the part of the measure which we want to
redistribute along which Entm is displacement convex. We have
1
1
Entm (Γ 1 ) ≤ Entm (Γ0 ) + Entm (Γ1 ) ≤ log M.
2
2
2
On the other hand, by Jensen’s inequality we always have
Z
Entm (Γ 1 ) =
ρ 1 log ρ 1 dm
2
2
E 2
Z
Z
1
≥ m(E ) − ρ 1 dm log − ρ 1 dm ≥ log
,
m(E )
E 2
E 2
where E = {x ∈ X : ρ 1 (x) > 0} and Γt = ρt m. Thus
2
m(E ) ≥
Tapio Rajala
1
.
M
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
This is why ||ρ1/2||∞ ≤ max{||ρ0 ||∞ , ||ρ1 ||∞ }.
The CD(K , ∞) condition gives a new well spread midpoint for the
high-density part of the old midpoint.
ρ0
ρ1
ρ1
2
M
M
M
A
X
E
X
X
E
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Taking a weighted combination of this new midpoint measure and
the old one lowers the entropy.
ρ1
ρ0
ρ1
2
M
M
X
M
X
Tapio Rajala
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Therefore at the minimum of the entropy among the midpoints we
have the density bound.
ρ1
ρ0
ρ1
2
M
M
X
M
X
Tapio Rajala
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
The rest of the geodesic.
When we continue taking minimizers in the next level midpoints
the bound is preserved.
ρ3
ρ0
ρ1
ρ1
ρ1
4
M
4
2
M
M
X
M
X
Tapio Rajala
M
X
X
X
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
The rest of the geodesic.
Finally we end up with a complete geodesic with the density bound.
ρ0
ρ1
M
t
X
X
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Question
MCP(K , N) ⇒ Local Poincaré?
Question
Does there exist optimal maps in CD(K , N)-spaces from every
µ ≪ m? (not all plans are given by maps)
Question
Local-to-global for CD(K , ∞)?
Question
Are RCD(K , ∞)-spaces non-branching?
Question
Are RCD(K , ∞)-spaces Ricci-limits?
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
J. Lott and C. Villani, Ricci curvature for metric-measure
spaces via optimal transport, Ann. of Math. 169 (2009), no. 3,
903–991.
K.-T. Sturm, On the geometry of metric measure spaces. I,
Acta Math. 196 (2006), no. 1, 65–131.
K.-T. Sturm, On the geometry of metric measure spaces. II,
Acta Math. 196 (2006), no. 1, 133–177.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
T. R., Local Poincaré inequalities from stable curvature
conditions on metric spaces, Calc. Var. Partial Differential
Equations, 44 (2012), 477–494.
T. R., Interpolated measures with bounded density in metric
spaces satisfying the curvature-dimension conditions of Sturm,
J. Funct. Anal., 263 (2012), no. 4, 896–924.
T. R., Improved geodesics for the reduced curvature-dimension
condition in branching metric spaces, Discrete Contin. Dyn.
Syst., 33 (2013), 3043–3056.
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
L. Ambrosio, N.Gigli and G. Savaré, Metric measure spaces
with Riemannian Ricci curvature bounded from below, preprint
(2011).
L. Ambrosio, N.Gigli, A. Mondino and T.R., Riemannian Ricci
curvature lower bounds in metric spaces with σ-finite measure,
Trans. Amer. Math. Soc., to appear.
N. Gigli, Optimal maps in non branching spaces with Ricci
curvature bounded from below, Geom. Funct. Anal. 22 (2012),
no. 4, 990–999.
T. R. and K.-Th. Sturm, Non-branching geodesics and optimal
maps in strong CD(K , ∞)-spaces, preprint (2012).
Tapio Rajala
CD(K , ∞)-spaces
RCD(K , ∞)-spaces
definition
Thank you!
Tapio Rajala

Branching geodesics in metric spaces with Ricci curvature

Transcription

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