Ricci curvature entropy and optimal transport Summer School in Grenoble

Description

Niveau: Supérieur, Doctorat, Bac+8
Ricci curvature, entropy and optimal transport – Summer School in Grenoble 2009 – ‘Optimal Transportation: Theory and Applications' Shin-ichi OHTA? Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, JAPAN (e-mail: ) These notes are the planned contents of my lectures. Some parts could be only briefly explained or skipped due to the lack of time or possible overlap with other lectures. The aim of these lectures is to review the recent development on the relation between optimal transport theory and Riemannian geometry. Ricci curvature is the key ingredient. Optimal transport theory provides a good characterization of lower Ricci curvature bounds without using differentiable structure. Then it can be considered as the ‘definition' of lower Ricci curvature bounds of metric measure spaces. In 1, we recall the definition of the Ricci curvature of a Riemannian manifold and the classical Bishop-Gromov volume comparison theorem. In 2, we start with Brunn- Minkowski inequalities in (weighted) Euclidean spaces, and show that a lower weighted Ricci curvature bound for a weighted Riemannian manifold is equivalent to some convexity inequality of entropy, called the curvature-dimension condition. In 3, we give the precise definition of the curvature-dimension condition for metric measure spaces, and see that it is stable under the measured Gromov-Hausdorff convergence.

• ric ≥


• ricci curvature


• relation between optimal


• riemannian manifold


• optimal transport


• bishop-gromov volume


• alexandrov spaces


• rectifiable curve ?

Sujets

• Grenoble
• Kyoto University
• Ricci curvature
• Riemannian manifold
• Transportation theory

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 28
• Langue: English

Extrait

Ricci curvature, entropy and optimal transport

– Summer School in Grenoble 2009 –
‘Optimal Transportation:Theory and Applications’

∗
Shin-ichi OHTA
Department of Mathematics, Faculty of Science, Kyoto University,
Kyoto 606-8502, JAPAN (e-mail:[e-mail])

These notes are the planned contents of my lectures.Some parts could be only briefly
explained or skipped due to the lack of time or possible overlap with other lectures.
The aim of these lectures is to review the recent development on the relation between
optimal transport theory and Riemannian geometry.Ricci curvature is the key ingredient.
Optimal transport theory provides a good characterization of lower Ricci curvature bounds
without using differentiable structure.Then it can be considered as the ‘definition’ of lower
Ricci curvature bounds of metric measure spaces.
In§1, we recall the definition of the Ricci curvature of a Riemannian manifold and
the classical Bishop-Gromov volume comparison theorem.In§2, we start with
BrunnMinkowski inequalities in (weighted) Euclidean spaces, and show that a lower weighted
Ricci curvature bound for a weighted Riemannian manifold is equivalent to some convexity
inequality of entropy, called the curvature-dimension condition.In§3, we give the precise
definition of the curvature-dimension condition for metric measure spaces, and see that
it is stable under the measured Gromov-Hausdorff convergence.§4 is devoted to some
geometric applications of the curvature-dimension condition.The final lecture will be
concerned with some of related topics summarized in§5. Althoughwe concentrate on
rather geometric aspects, these lectures will be far from exhaustive.Interested readers
can find more references in Further Reading at the end of each section (except§5).

0

Notations

First of all, we collect some notations we use for convenience.Throughout these lectures,
(M, g) is ann-dimensional complete Riemannian manifold without boundary withn≥2,
volgstands for the Riemannian volume measure ofg.
A metric space is called ageodesic spaceif any two pointsx, y∈Xcan be connected
by a rectifiable curveγ: [0,1]−→Xof lengthd(x, y) withγ(0) =xandγ(1) =y. Such
minimizing curves parametrized proportionally to arc length are calledminimal geodesics.
Open and closed balls of centerxand radiusrwill be denoted byB(x, r) andB(x, r). A

∗
Partly supported by the Grant-in-Aid for Young Scientists (B) 20740036.

1

metric measure spacewill be a triple (X, d, m) consisting of a geodesic space (X, d) and
a Borel measuremon it such that 0< m(B(x, r))<∞for allx∈Xand 0< r <∞.
For a metric space (X, d), we denote byP(X) the set of Borel probability measures
onX, byP2(X)⊂ P(X) the subset consisting of measures of finite second moment, and
W
byPc(X)⊂ P2(X) the set of compactly supported measures.Thendstands for the
2
2
L-Wasserstein distance onP2(X).
As usual in comparison geometry, the following functions will frequently appear in our
√
discussions. ForK∈R,N∈(1,∞) and 0< r(< π(N−1)/KifK >0), we define
 √√
(N−1)/Ksin(r K/(N−1)) ifK >0,

sK,N(r) :=rifK= 0,
 √√

−(N−1)/Ksinh(r−K/(N−1)) ifK <0.

In addition, fort∈(0,1), we set
( )N−1
(tr)
tsK,N
β(r) :=,
K,N
tsK,N(r)

1

Ricci curvature

2 2
t K(1−t)r /6
β(r) :=e .
K,∞

Take a vector fieldJalong a geodesicγ: [0,1]−→M. IfJis the variational vector field
of some family of geodesics, thenJis called aJacobi field. Jacobifields satisfy theJacobi
equation
2
D J+R(J,˙γ) ˙γ= 0,(1.1)
˙γ
whereR:T M⊗T M⊗T M−→T Mis the curvature tensor determined by the Riemannian
metricglinearly independent tangent vectors. Forv, w∈TxM,

hR(w, v)v, wi
K(v, w) :=
2 22
|v| |wh| −v, wi

is thesectional curvatureof the 2-plane spanned byvandwa unit vector. Forv∈TxM,
theRicci curvatureofvis defined as the trace ofK(v,∙):

n−1
∑
Ric(v) :=K(v, ei),
i=1

n
where{e}n ort
i i=1is ahonormal basis ofTxMwithen=v.
The sectional curvatureKnaturally controls the behavior of (especially, the second
order derivative of) the distance along geodesics.For instance, the lower boundK ≥kfor
k∈Ris equivalent to that every geodesic triangle inMis ‘thicker’ than the triangle with
the same side lengths in the two-dimesional space form of constant sectional curvature
ktriangle comparison condition makes sense also in metric spaces.. ThisSuch spaces
are calledAlexandrov spaces, and deeply investigated from the geometric and analytic
viewpoints.

2

Since it is taking the trace, the Ricci curvature has less information and controls only
the behavior of the measurem= volg. Alonga unit speed geodesicγ: [0, l]−→M,
n−1
consider Jacobi fields{Ji}alongγgiven by
i=1

Ji(t) :=D(expγ(0))tγ˙ (0)(tei)∈Tγ(t)M

n
w
using an orthonormal basis{ei}i=1ithen= ˙γ((0). Definen−1)×(n−1) matrix-valued
functions
1
′ −1
A:= (hJi, Jji),U:=A A,R:= (hR(Ji,˙γ)γ˙, Jji).
2
ThenUis symmetric and satisfies the (matrix)Riccati equation

′2−1
U+U+RA= 0.

Taking the trace yields
′2
(trUtr() +U) + Ric(γ˙ )= 0
2 2
which with tr(U)≥(trU)/(n−1) shows

(1.2)

2
(trU)
′
(trU) ++ Ric(γ˙ )≤0.(1.3)
n−1
This estimate implies (a version of) theBishop comparison theorem
2[ ]
dRic(γ˙ )
1/2(n−1) 1/2(n−1)
(detA)≤ −(detA).(1.4)
2
dt n−1
Now we assume Ric≥Kand by integrating (1.4) find theBishop-Gromov volume
comparison theorem
∫
R
n−1
m(B(x, R))sK,n(t)dt
0
∫
≤r(1.5)
n−1
m(B(x, r))sK,n(t)dt
0
√
for anyx∈Mand 0< r < R(≤π(n−1)/KifK >0).
The Bishop and Bishop-Gromov comparison theorems give us a nice intuition how
spaces with lower Ricci curvature bounds look like.Although bounding Ricci curvature
from below is essential in many analytic applications, how to characterize such spaces
without using differentiable structure had been a long standing important problem.An
answer to this question is the topic of§2.

Further readingSee [Ch] for comparison theorems in Riemannian geometry.Basic
references of Alexandrov spaces are [BGP], [OtS] and [BBI].A property corresponding to
the Bishop comparison theorem (1.4) was proposed as lower Ricci curvature bounds for
metric measure spaces by Cheeger and Colding [CC] (as well as Gromov [Gr]), and used
to study the limit spaces of Riemannian manifolds with uniform lower Ricci curvature
bounds. However,its systematic investigation has not been done until [Oh1] and [St4]
(see also [KS1] and [St1] for related antecedents).

3

2

The curvature-dimension condition

n
The classicalBrunn-Minkowski inequalityin the Euclidean spaceRasserts the concavity
of then-th root of the Lebesgue measure:
( )1/n
1/n1/n
mL(1−t)A+tB≥(1−t)mL(A) +tmL(B) (2.1)

n
fort∈[0,1] and measurable setsA, B⊂R, where

(1−t)A+tB:={(1−t)x+ty|x∈A, y∈B}.

We can prove (2.1) using optimal transport between uniform distributions onAandB,
the key ingredient is the inequality of arithmetic and geometric means
[ ()]1/n
1/n
det (1−t)In+tA ≥(1−t) +t(detA)

for ann×nsymmetric matrixA. Morecareful argument shows that a weighted Euclidean
n−ψ∞n
space (R, m=e mL) for someψ∈C(R) satisfies a generalization of the
BrunnMinkowski inequality
( )1/N
1/N1/N
m(1−t)A+tB≥(1−t)m(A) +tm(B) (2.2)

forN∈(n,∞) if (and only if)

2
hgradψ, vi
Hessψ(v, v)− ≥0 (2.3)
N−n
n
holds for all (unit) vectorsv∈TR.
A quantity corresponding to (2.3) is called theweighted Ricci curvaturein the theory
−ψ∞
of weighted Riemannian manifolds (M, g, m=evolg) withψ∈C(M):

2
hgradψ, vi
RicN(v) := Ric(v) + Hessψ(v, v)−(2.4)
N−n
for unit tangent vectorsv∈T Minfinite dimensional case (. TheN=∞) amounts to the
Bakry-´merytensor
Ric∞(v) := Ric(v) + Hessψ(v, v).(2.5)
We also define Ricn(v) := Ric(v) ifhgradψ, vi= 0, and Ricn(v) :=−∞otherwise. Recall
that the Bishop-Gromov volume comparison (1.5) with Ric≥0 yields
( )n
m(B(x, R))R
≤
m(B(x, r))r

which can be regarded as the Brunn-Minkowski inequality between{x}andB(x, R) with
t=r/Rit is natural to expect that lower Ricci curvature bounds relate to some. Therefore
interpolation inequalities like the Brunn-Minkowski inequality.The curvature-dimension
conditionCD(K, N) is actually a generalization of the Brunn-Minkowski inequality to

4

pairs of (not necessarily uniformly distributed) probability measures.The precise
definition ofCD(K, N) will be given in§3, here we see that the core inequality ofCD(K, N) is
equivalent to RicN≥Kfor weighted Riemannian manifolds.
GivenN∈[n,∞) and absolutely continuous probability measureµ=ρm∈ P(M),
we define theR´nyi entropyas
∫
1−1/N
SN(µ) :=−ρ dm.(2.6)
M
We also define therelative entropyby
∫
Ent(µ) :=ρlogρ dm.(2.7)
M
−ψ
Theorem 2.1A weighted Riemannian manifold(M, g, m=evolg)satisfiesRicN≥K
for someK∈RandN∈[n,∞)if and only if any pair of absolutely continuous probability
measuresµ0=ρ0m,µ1=ρ1m∈ Pc(M)satisfies
∫
( )1/N
1−t−1/N
SN(µt)≤ −(1−t)β d(x, y)ρ0(x)dπ(x, y)
K,N
M×M
∫
( )1/N
t−1/N
d(x, y)x, y),(2.8)
−t βK,