Ricci curvature, entropy and optimal transport
Shin-ichi OHTA∗
Department of Mathematics, Faculty of Science, Kyoto University,
Kyoto 606-8502, JAPAN (e-mail: sohta@math.kyoto-u.ac.jp)

Abstract
This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimaltransport in the space of probability measures characterizes lower bounds of the
Ricci curvature. We then discuss geometric properties of general metric measurespaces satisfying this convexity condition.
Mathematics Subject Classiﬁcation (2000): 53C21, 53C23, 53C60, 28A33,
28D20
Keywords: Ricci curvature, entropy, optimal transport, curvature-dimension condition

1

Introduction

This article is extended notes based on the author’s lecture series in summer school atUniversité Joseph Fourier, Grenoble: ‘Optimal Transportation: Theory and Applications’.
The aim of these ﬁve lectures (corresponding to Sections 3–7) was to review the recent
impressive development on the interplay between optimal transport theory and Riemannian geometry. Ricci curvature and entropy are the key ingredients. See [Lo2] for a survey
in the same spirit with a slightly diﬀerent selection of topics.
Optimal transport theory is concerned with the behavior of transport between two
probability measures in a metric space. We say that such transport is optimal if it minimizes a certain cost function typically deﬁned from the distance of the metric space.
Optimal transport naturally inherits the geometric structure of the underlying space,especially Ricci curvature plays a crucial role for describing optimal transport in Riemannian manifolds. In fact, optimal transport is always performed along geodesics, and we
obtain Jacobi ﬁelds as their variational vector ﬁelds. The behavior of these Jacobi ﬁelds
is controlled by the Ricci curvature as is usual in comparison geometry. In this way, alower Ricci curvature bound turns out to be equivalent to a certain convexity property ofentropy in the space of probability measures. The latter convexity condition is called thecurvature-dimension condition, and it can be formulated without using the diﬀerentiable∗

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

1

structure. Therefore the curvature-dimension condition can be regarded as a ‘deﬁnition’of a lower Ricci curvature bound for general metric measure spaces, and implies manyanalogous properties in an interesting way.
A prerequisite is the basic knowledge of optimal transport theory and Wassersteingeometry. Riemannian geometry is also necessary in Sections 3, 4, and is helpful for
better understanding of the other sections. We refer to [AGS], [Vi1], [Vi2] and otherarticles in this proceeding for optimal transport theory, [CE], [Ch] and [Sak] for thebasics of (comparison) Riemannian geometry. We discuss Finsler geometry in Section 7,
for which we refer to [BCS], [Sh2] and [Oh5]. Besides them, main references are [CMS1],[CMS2], [vRS], [St3], [St4], [LV2], [LV1] and [Vi2, Chapter III].
The organization of this article is as follows. After summarizing some notations weuse, Section 3 is devoted to the deﬁnition of the Ricci curvature of Riemannian manifolds
and to the classical Bishop-Gromov volume comparison theorem. In Section 4, we startwith the Brunn-Minkowski inequalities in (unweighted or weighted) Euclidean spaces,and explain the equivalence between a lower (weighted) Ricci curvature bound for a(weighted) Riemannian manifold and the curvature-dimension condition. In Section 5, wegive the precise deﬁnition of the curvature-dimension condition for metric measure spaces,and see that it is stable under the measured Gromov-Hausdorﬀ convergence. Section 6
is concerned with several geometric applications of the curvature-dimension condition
followed by related open questions. In Section 7, we verify that this kind of machinery
is useful also in Finsler geometry. We ﬁnally discuss three related topics in Section 8.
Interested readers can ﬁnd more references in Further Reading at the end of each section(except the last section).
Some subjects in this article are more comprehensively discussed in [Vi2, Part III].
Despite these inevitable overlaps with Villani’s massive book, we try to argue in a moregeometric way, and mention recent development. Analytic applications of the curvaturedimension condition are not dealt with in these notes, for which we refer to [LV1], [LV2]and [Vi2, Chapter III] among others.
I would like to express my gratitude to the organizers for the kind invitation to thefascinating summer school, and to all the audience for their attendance and interest. I
also thank the referee for careful reading and valuable suggestions.

2

NotationsThroughout the article except Section 7, (M, g) is an n-dimensional, connected, complete
C ∞ -Riemannian manifold without boundary such that n ≥ 2, volg stands for the Riemannian volume measure of g. A weighted Riemannian manifold (M, g, m) will mean a
Riemannian manifold (M, g) endowed with a conformal deformation m = e−ψ volg of volgwith ψ ∈ C ∞ (M ). Similarly, a weighted Euclidean space (Rn , k · k, m) will be a Euclidean
space with a measure m = e−ψ voln , where voln stands for the n-dimensional Lebesgue
measure.
A metric space is called a geodesic space if any two points x, y ∈ X can be connectedby a rectiﬁable curve γ : [0, 1] −→ X of length d(x, y) with γ(0) = x and γ(1) = y. Such
minimizing curves parametrized proportionally to arc length are called minimal geodesics.
2

The open ball of center x and radius r will be denoted by B(x, r). We remark that, thanks
to the Hopf-Rinow theorem (cf. [Bal, Theorem 2.4]), a complete, locally compact geodesic
space is proper, i.e., every bounded closed set is compact.
In this article, we mean by a metric measure space a triple (X, d, m) consisting of acomplete, separable geodesic space (X, d) and a Borel measure m on it. Our deﬁnitionof the curvature-dimension condition will include the additional (but natural) condition
that 0 < m(B(x, r)) < ∞ holds for all x ∈ X and 0 < r < ∞. We extend m to an outermeasure in the Brunn-Minkowski inequalities (Theorems 4.1, 4.3, 6.1, see Remark 4.2 formore details).
For a complete, separable metric space (X, d), P(X) stands for the set of Borel probability measures∫ on X. Deﬁne P2 (X) ⊂ P(X) as the set of measures of ﬁnite second
moment (i.e., X d(x, y)2 dµ(y) < ∞ for some (and hence all) x ∈ X). We denote byPb (X) ⊂ P2 (X), Pc (X) ⊂ Pb (X) the sets of measures of bounded or compact support,
respectively. Given a measure m on X, denote by P ac (X, m) ⊂ P(X) the set of abso2
lutely continuous measures with respect to m. Then dW
2 stands for the L -(KantorovichRubinstein-)Wasserstein distance of P2 (X). The push-forward of a measure µ by a map
F will be written as F] µ.
As usual in comparison geometry, the following functions
will frequently appear in our√
discussions. For K ∈ R, N ∈ (1, ∞) and 0 < r (< π (N − 1)/K if K > 0), we set √√(N − 1)/K sin(r K/(N − 1))
if K > 0,r
if K = 0,
sK,N (r) :=
(2.1)

√
 √
−(N − 1)/K sinh(r −K/(N − 1)) if K < 0.
This is the solution to the diﬀerential equation
s00K,N +K
sK,N = 0
N −1

(2.2)

with the initial conditions sK,N (0) = 0 and s0K,N (0) = 1. For n ∈ N with n ≥ 2, sK,n (r)n−1
is proportional to the area of the sphere of radius r in the n-dimensional space form of
constant sectional curvature K/(n − 1) (see Theorem 3.2 and the paragraph after it). In
addition, using sK,N , we deﬁne)N −1(
sK,N (tr)2 2ttβK,N (r) :=,βK,∞(r) := eK(1−t )r /6(2.3)
tsK,N (r)for K, N, r as above and t ∈ (0, 1). This function plays a vital role in the key inﬁnitesimalinequality (4.14) of the curvature-dimension condition.

3

Ricci curvature and comparison theorems

We begin with the basic concepts of curvature in Riemannian geometry and several comparison theorems involving lower bounds of the Ricci curvature. Instead of giving thedetailed deﬁnition, we intend to explain the geometric intuition of the sectional and Ricci
curvatures through comparison geometry.
3

Curvature is one of the most important quantities in Riemannian geometry. By puttingsome conditions on the value of the (sectional or Ricci) curvature, we obtain various quantitative and qualitative controls of distance, measure, geodesics and so forth. Comparison
geometry is speciﬁcally interested in spaces whose curvature is bounded by a constantfrom above or below. In other words, we consider a space which is more positively or
negatively curved than a space form of constant curvature, and compare these spaces
from various viewpoints.
The n-dimensional (simply connected) space form Mn (k) of constant sectional curvature k ∈ R is the unit sphere Sn for k = 1; the Euclidean space Rn for k = 0; andthe hyperbolic space Hn for k = −1. Scaling√gives general space forms for all k ∈ R,
e.g., Mn (k) for k > 0 is the sphere of radius 1/ k in Rn+1 with the induced Riemannian
metric.

3.1

Sectional curvature

Given linearly independent tangent vectors v, w ∈ Tx M , the sectional curvature K(v, w) ∈R reﬂects the asymptotic behavior of the distance function d(γ(t), η(t)) near t = 0 between
geodesics γ(t) = expx (tv) and η(t) = expx (tw). That is to say, the asymptotic behaviorof d(t) := d(γ(t), η(t)) near t = 0 is same as the distance between geodesics, with thesame speed and angle between them, in the space form of curvature k = K(v, w). (SeeFigure 1 which represents isometric embeddings of γ and η into R2 such that d(γ(t), η(t))
coincides with the Euclidean distance.)

γ

η

γ

η

γ

η

x

x

x

K>0

K=0

K<0

Figure 1
Assuming kvk = kwk = 1 for simplicity, we can compute d(t) in the space form Mn (k)by using the spherical/Euclidean/hyperbolic law of cosines as (cf. [Sak, Section IV.1])√√(√)
cos kd(t) = cos2 ( kt) + sin2 ( kt) cos ∠(v, w)
for k > 0,
2
2
2
d(t)
=
2t
−
2t
cos
∠(v,
w)
for
k = 0,√√)(√
2
2
cosh −kd(t) = cosh ( −kt) − sinh ( −kt) cos ∠(v, w) for k < 0.

4

Observe that the dimension n does not appear in these formulas. The sectional curvatureK(v, w) depends only on the 2-plane (in Tx M ) spanned by v and w, and coincides with
the Gaussian curvature at x if n = 2.
More precise relation between curvature and geodesics can be described through Jacobi
ﬁelds. A C ∞ -vector ﬁeld J along a geodesic γ : [0, l] −→ M is called a Jacobi ﬁeld if itsolves the Jacobi equation()
Dγ̇ Dγ̇ J(t) + R J(t), γ̇(t) γ̇(t) = 0(3.1)
for all t ∈ [0, l]. Here Dγ̇ denotes the covariant derivative along γ, and R : Tx M ⊗Tx M −→ Tx∗ M ⊗ Tx M is the curvature tensor determined by the Riemannian metricg. Another equivalent way of introducing a Jacobi ﬁeld is to deﬁne it as the variational
vector ﬁeld J(t) = (∂σ/∂s)(0, t) of some C ∞ -variation σ : (−ε, ε) × [0, l] −→ M suchthat σ(0, t) = γ(t) and that every σs := σ(s, ·) is geodesic. (This characterization of
Jacobi ﬁelds needs only the class of geodesics, and then it is possible to regard (3.1) as
the deﬁnition of R.) For linearly independent vectors v, w ∈ Tx M , the precise deﬁnitionof the sectional curvature isK(v, w) :=hR(w, v)v, wi.
kvk2 kwk2 − hv, wi2

It might be helpful to compare (3.1) with (2.2).
Remark 3.1 (Alexandrov spaces) Although it is not our main subject, we brieﬂycomment on comparison geometry involving lower bounds of the sectional curvature. As
the sectional curvature is deﬁned for each two-dimensional subspace in tangent spaces,
it controls the behavior of two-dimensional subsets in M , in particular, triangles. The
classical Alexandrov-Toponogov comparison theorem asserts that K ≥ k holds for somek ∈ R if and only if every geodesic triangle in M is thicker than the triangle withthe same side lengths in M2 (k). See Figure 2 for more details, where M is of K ≥ k,and then d(x, w) ≥ d(x̃, w̃) holds between geodesic triangles with the same side lengths(d(x, y) = d(x̃, ỹ), d(y, z) = d(ỹ, z̃), d(z, x) = d(z̃, x̃)) as well as d(y, w) = d(ỹ, w̃).
y

w

z

ỹ

w̃

z̃

M⊃

⊂ M2 (k)xd(x, w) ≥ d(x̃, w̃)Figure 25x̃The point is that we can forget about the dimension of M , because the sectional
curvature cares only two-dimensional subsets. The above triangle comparison property is
written by using only distance and geodesics, so that it can be formulated for metric spaces
having enough geodesics (i.e., geodesic spaces). Such spaces are called Alexandrov spaces,
and there are deep geometric and analytic theories on them (see [BGP], [OtS], [BBI,
Chapters 4, 10]). We discuss optimal transport and Wasserstein geometry on Alexandrov
spaces in Subsection 8.2.

3.2

Ricci curvature

Given a unit vector v ∈ Tx M , we deﬁne the Ricci curvature of v as the trace of thesectional curvature K(v, ·),n−1∑
Ric(v) :=K(v, ei ),i=1where {ei }n−1i=1 ∪ {v} is an orthonormal basis of Tx M .

We will mean by Ric ≥ K forK ∈ R that Ric(v) ≥ K holds for all unit vectors v ∈ T M . As we discussed in theprevious subsection, sectional curvature controls geodesics and distance. Ricci curvature
has less information since we take the trace, and naturally controls the behavior of themeasure volg .
The following is one of the most important theorems in comparison Riemannian geometry, that asserts that a lower bound of the Ricci curvature implies an upper boundof the volume growth. The proof is done via calculations involving Jacobi ﬁelds. Recall(2.1) for the deﬁnition of the function sK,n .
Theorem 3.2 (Bishop-Gromov volume comparison) Assume that Ric√ ≥ K holds
for some K ∈ R. Then we have, for any x ∈ M and 0 < r < R (≤ π (n − 1)/K if
K > 0),∫R
sK,n (t)n−1 dt
volg (B(x, R))
≤ ∫0r
.
(3.2)
n−1 dtvolg (B(x, r))
s
(t)
K,n
0
Proof. Given a unit vector v ∈ Tx M , we ﬁx a unit speed minimal geodesic γ : [0, l] −→ Mwith γ̇(0) = v and an orthonormal basis {ei }n−1i=1 ∪ {v} of Tx M . Then we consider thevariation σi : (−ε, ε)×[0, l] −→ M deﬁned by σi (s, t) := expx (tv+stei ) for i = 1, . . . , n−1,
and introduce the Jacobi ﬁelds {Ji }n−1i=1 along γ given byJi (t) :=∂σi(0, t) = D(expx )tv (tei ) ∈ Tγ(t) M∂s(see Figure 3, where s > 0).

6

Ji
6

6
6

x

6

σi (s, ·)γ

Figure 3
Note that Ji (0) = 0, Dγ̇ Ji (0) = ei , hJi , γ̇i ≡ 0 (by the Gauss lemma) and hDγ̇ Ji , γ̇i ≡ 0(by (3.1) and hR(Ji , γ̇)γ̇, γ̇i ≡ 0). We also remark that γ(t) is not conjugate to x for allt ∈ (0, l) (and hence {Ji (t)}n−1
Hence wei=1 ∪ {γ̇(t)} is a basis of Tγ(t) M ) since γ is minimal.
∑n−1
n−1
ﬁnd an (n − 1) × (n − 1) matrix U(t) = (uij (t))i,j=1 such that Dγ̇ Ji (t) = j=1 uij (t)Jj (t)for t ∈ (0, l). We deﬁne two more (n − 1) × (n − 1) matrices
(­ (
)
®)n−1
(
)n−1
A(t) := hJi (t), Jj (t)i i,j=1 , R(t) := R Ji (t), γ̇(t) γ̇(t), Jj (t).
i,j=1Note that A and R are symmetric matrices. Moreover, we have tr(R(t)A(t)−1 ) =Ric(γ̇(t)) as A(t) is the matrix representation of the metric g in the basis {Ji (t)}n−1
i=1 of the⊥orthogonal complement γ̇(t)of
γ̇(t).
Tobeprecise,choosing
an
(n
−
1)
×
(n
−
1) matrix
∑n−1
n−1
n−1
C = (cij )i,j=1 such that { j=1 cij Jj (t)}i=1 is orthonormal, we observe In = CA(t)C t (C tis the transpose of C) and
n−1
∑
­ (
)
®
(
)
(
)
R cij Jj (t), γ̇(t) γ̇(t), cik Jk (t) = tr CR(t)C t
Ric γ̇(t) =i,j,k=1()= tr R(t)A(t)−1 .
Claim 3.3 (a) It holds that UA = AU t . In particular, we have 2U = A0 A−1 .
(b) The matrix U is symmetric and we have tr(U 2 ) ≥ (tr U)2 /(n − 1).
Proof. (a) The ﬁrst assertion easily follows from the Jacobi equation (3.1) and the symmetry of R, indeed,d{hDγ̇ Ji , Jj i − hJi , Dγ̇ Jj i} = hDγ̇ Dγ̇ Ji , Jj i − hJi , Dγ̇ Dγ̇ Jj idt
= −hR(Ji , γ̇)γ̇, Jj i + hJi , R(Jj , γ̇)γ̇i = 0.
Thus we have A0 = UA + AU t = 2UA which shows the second assertion.
(b) Recall that∂σi
∂σi(0, t) = γ̇(t),(0, t) = Ji (t)∂t∂shold for t ∈ (0, l). As [∂/∂s, ∂/∂t] = 0, we have
(
)
(
)
∂σi
∂σi
Dγ̇ Ji (t) = Dt
(0, t) = Ds
(0, t).
∂s∂t
7

Now, we introduce the function({
f : expx

tv +

n−1
∑

})
¯
¯
si tei ¯ t ∈ [0, l], |si | < ε
−→ R

i=1

so that f (expx (tv +∑n−1

i=1 si tei )) = t. We derive from ∇f (σi (s, t)) = (∂σi /∂t)(s, t) that

(

Ds

)
(
)
(
)
∂σi
(0, t) = DJi (∇f ) γ(t) = ∇2 f Ji (t) ,∂twhere h∇2 f (w), w0 i = Hess f (w, w0 ). This means that U is the matrix presentation ofn−1
the symmetric form ∇2 f (restricted in γ̇ ⊥ ) with respect to the basis {Ji }i=1
. Therefore
U is symmetric. By denoting the eigenvalues of U by λ1 , . . . , λn−1 , the Cauchy-Schwarz
inequality shows that(tr U) =
2

(∑
n−1

)2
λi

≤ (n − 1)

i=1

n−1
∑

λ2i = (n − 1) tr(U 2 ).

i=1♦
We calculate, by using Claim 3.3(a),[]0(detA)1/2(n−1) =1(detA)1/2(n−1)−1 · detA tr(A0 A−1 )2(n − 1)1=
(detA)1/2(n−1) tr U.
n−1

Then Claim 3.3(b) yields[]00(detA)1/2(n−1) =1
1
1/2(n−1)
2(detA)(trU)+(detA)1/2(n−1) tr(U 0 )(n − 1)2n−1
1≤(detA)1/2(n−1) {tr(U 2 ) + tr(U 0 )}.
n−1

We also deduce from Claim 3.3 and (3.1) that1
1
1
U 0 = A00 A−1 − (A0 A−1 )2 = (−2R + 2UAU )A−1 − 2U 2 = −RA−1 − U 2 .
2
22This implies the (matrix) Riccati equation
U 0 + U 2 + RA−1 = 0.
Taking the trace gives(tr U)0 + tr(U 2 ) + Ric(γ̇) = 0.

Thus we obtain from our hypothesis Ric ≥ K the diﬀerential inequality[]00
K(detA)1/2(n−1) ≤ −(detA)1/2(n−1) .
n−1
8

(3.3)

This is a version of the fundamental Bishop comparison theorem which plays a prominentrole in comparison geometry. Comparing (3.3) with (2.2), we have
}
]0
d {[
(detA)1/2(n−1) sK,n − (detA)1/2(n−1) s0K,ndt []00= (detA)1/2(n−1) sK,n − (detA)1/2(n−1) s00K,n ≤ 0,√
and hence (detA)1/2(n−1) /sK,n is non-increasing. Then integrating detA in unit vectors
v ∈ Tx M implies the area comparison theorem
areag (S(x, R))
sK,n (R)n−1≤,
areag (S(x, r))sK,n (r)n−1(3.4)

where S(x, r) := {y ∈ M | d(x, y) = r} and areag stands for the (n − 1)-dimensional
Hausdorﬀ measure associated with g (in other words, the volume measure of the (n − 1)dimensional Riemannian metric of S(x, r) induced from g).
Now, we integrate (3.4) in the radial direction. Set A(t) := areag (S(x, t)) and S(t) :=sK,n (t)n−1 , and recall that A/S is non-increasing. Hence we obtain the key inequality
∫ r
∫ R
∫
∫ R
∫ r
∫ R
A(r) rA dt
S dt ≥S dt
S dt ≥
S dt
A dt.
(3.5)
S(r) 00rr
0r
From here to the desired estimate (3.2) is the easy calculation as follows
∫
∫ r
∫ R
∫ r
∫ r
(
) R
n−1
volg B(x, r)
sK,n (t)dt =A dt
S dt +
A dt
S dt
0
0
r
0
0
∫ r
∫ R
∫ r
∫ r
∫
(
) r
≥
S dt
A dt +A dt
S dt = volg B(x, R)
sK,n (t)n−1 dt.
0

r

0

0

0

2
The sphere of radius r in the space form Mn (k) has area
an s(n−1)k,n (r)n−1 , where an∫ris the area of Sn−1 , and the ball of radius r has volume an 0 s(n−1)k,n (t)n−1 dt. Thus theright-hand side of (3.2) ((3.4), respectively) coincides with the ratio of the volume of balls(the area of spheres, respectively) of radius R and r in Mn (K/(n − 1)).
Theorem 3.2 √for K > 0 immediately implies a diameter bound. This ensures that thecondition R ≤ π (n − 1)/K in Theorem 3.2 is natural.
Corollary 3.4 (Bonnet-Myers diameter bound) If Ric ≥ K > 0, then we have√n−1diam M ≤ π
.
(3.6)
K
√
Proof. Put R := π (n − 1)/K and assume diam M ≥ R. Given x ∈ M , Theorem 3.2
implies that{}volg (B(x, R) \ B(x, R − ε))volg (B(x, R − ε))lim sup= lim sup 1 −volg (B(x, R))
volg (B(x, R))
ε↓0
ε↓0
∫R
s (t)n−1 dt
R−ε K,n= 0.
≤ lim sup ∫ R
n−1 dt
ε↓0
s
(t)
K,n
0
9

This shows areag (S(x, R)) = 0 and hence diam M ≤ R. To be precise, it follows fromareag (S(x, R)) = 0 that every point in S(x, R) must be a conjugate point of x. Therefore any geodesic emanating from x is not minimal after passing through S(x, R), andhence diam M = R. (A more direct proof in terms of metric geometry can be found in
Theorem 6.5(i).)
2nThe bound√ (3.6) is sharp, and equality is achieved only by the sphere M (K/(n − 1))of radius (n − 1)/K (compare this with Theorem 6.6).
As we mentioned in Remark 3.1, lower sectional curvature bounds are characterizedby simple triangle comparison properties involving only distance, and there is a successful
theory of metric spaces satisfying them. Then it is natural to ask the following question.

Question 3.5 How to characterize lower Ricci curvature bounds without using diﬀerentiable structure?
This had been a long standing important question, and we will see an answer in thenext section (Theorem 4.6). Such a condition naturally involves measure and dimension
besides distance, and should be preserved under the convergence of metric measure spaces(see Section 5).
Further Reading See, for instances, [CE], [Ch] and [Sak] for the fundamentals of Riemannian geometry and comparison theorems. A property corresponding to the Bishop
comparison theorem (3.3) was proposed as a lower Ricci curvature bound for metric measure spaces by Cheeger and Colding [CC] (as well as Gromov [Gr]), and used to study thelimit spaces of Riemannian manifolds with uniform lower Ricci curvature bounds. The
deep theory of such limit spaces is one of the main motivations for asking Question 3.5,
so that the stability deserves a particular interest (see Section 5 for more details). The
systematic investigation of (3.3) in metric measure spaces has not been done until [Oh1]and [St4] where we call this property the measure contraction property. We will revisit
this in Subsection 8.3. Here we only remark that the measure contraction property isstrictly weaker than the curvature-dimension condition.

4

A characterization of lower Ricci curvature boundvia optimal transport

The Bishop-Gromov volume comparison theorem (Theorem 3.2) can be regarded as aconcavity estimate of vol1/nalong the contraction of the ball B(x, R) to its center x.
gThis is generalized to optimal transport between pairs of uniform distributions (theBrunn-Minkowski inequality) and, moreover, pairs of probability measures (the curvaturedimension condition). Figure 4 represents the diﬀerence between contraction and transport (see also Figures 5, 8).

10

?
-

z
:

¾
6

Bishop-Gromov

Brunn-Minkowski/Curvature-dimension
Figure 4

The main theorem in this section is Theorem 4.6 which asserts that, on a weighted
Riemannian manifold, the curvature-dimension condition is equivalent to a lower boundof the corresponding weighted Ricci curvature. In order to avoid lengthy calculations,
we begin with Euclidean spaces with or without weight, and see the relation between
the Brunn-Minkowski inequality and the weighted Ricci curvature. Then the general
Riemannian situation is only brieﬂy explained. We hope that our simpliﬁed argument
will help the readers to catch the idea of the curvature-dimension condition.

4.1

Brunn-Minkowski inequalities in Euclidean spaces

For later convenience, we explain fundamental facts of optimal transport theory on Euclidean spaces. Given µ0 , µ1 ∈ Pc (Rn ) with µ0 ∈ P ac (Rn , voln ), there is a convex function
f : Rn −→ R such that the map
Ft (x) := (1 − t)x + t∇f (x),t ∈ [0, 1],

gives the unique optimal transport from µ0 to µ1 (Brenier’s theorem, [Br]). Precisely,t 7−→ µt := (Ft )] µ0 is the unique minimal geodesic from µ0 to µ1 with respect to the
L2 -Wasserstein distance. We remark that the convex function f is twice diﬀerentiable
a.e. (Alexandrov’s theorem, cf. [Vi2, Chapter 14]). Thus ∇f makes sense and Ft is
diﬀerentiable a.e. Moreover, the Monge-Ampère equation
(
) (
)
(4.1)
ρ1 F1 (x) det DF1 (x) = ρ0 (x)holds for µ0 -a.e. x.
Now, we are ready for proving the classical Brunn-Minkowski inequality in the (unweighted) Euclidean space (Rn , k · k, voln ). Brieﬂy speaking, it asserts that vol1/n
isnconcave. We shall give a proof based on optimal transport theory. Given two (nonempty)
sets A, B ⊂ Rn and t ∈ [0, 1], we set(1 − t)A + tB := {(1 − t)x + ty | x ∈ A, y ∈ B}.
(See Figure 5, where (1/2)A + (1/2)B has much more measure than A and B.)
11AB(1/2)A + (1/2)B
Figure 5
Theorem 4.1 (Brunn-Minkowski inequality) For any measurable sets A, B ⊂ Rn
and t ∈ [0, 1], we have(
)1/nvoln (1 − t)A + tB
≥ (1 − t) voln (A)1/n + t voln (B)1/n .
(4.2)
Proof. We can assume that both A and B are bounded and of positive measure. The
case of voln (A) = 0 is easily checked by choosing a point x ∈ A, as we have
(
)1/n
(
)1/n
voln (1 − t)A + tB
≥ voln (1 − t){x} + tB
= t voln (B)1/n .
If either A or B is unbounded, then applying (4.2) to bounded sets yields()1/nvoln (1 − t){A ∩ B(0, R)} + t{B ∩ B(0, R)}
(
)1/n
(
)1/n
≥ (1 − t) voln A ∩ B(0, R)+ t voln B ∩ B(0, R).
We take the limit as R go to inﬁnity and obtain()1/nvoln (1 − t)A + tB
≥ (1 − t) voln (A)1/n + t voln (B)1/n .
Consider the uniform distributions on A and B,µ0 = ρ0 voln :=χAvoln ,
voln (A)µ1 = ρ1 voln :=χBvoln ,
voln (B)where χA stands for the characteristic function of A. As µ0 is absolutely continuous, there
is a convex function f : Rn −→ R such that the map F1 = (1 − t) IdRn +t∇f , t ∈ [0, 1],
is the unique optimal transport from µ0 to µ1 . Between the uniform distributions µ0 andµ1 , the Monge-Ampère equation (4.1) simply means thatdet(DF1 ) =voln (B)voln (A)µ0 -a.e.
Note that DF1 = Hess f is symmetric and positive deﬁnite µ0 -a.e., since f is convex
and det(DF1 ) > 0. We shall estimate det(DFt ) = det((1 − t)In + tDF1 ) from above
12and below. To do so, we denote the eigenvalues of DF1 by λ1 , . . . , λn > 0 and apply theinequality of arithmetic and geometric means to see{}1/n {}1/n(1 − t)ntn det(DF1 )+det((1 − t)In + tDF1 )det((1 − t)In + tDF1 ){∏}1/n { ∏
}1/n
n
n
1−t
tλi
=
+
(1 − t) + tλi(1 − t) + tλi
i=1
i=1
{
}
n
tλi
1∑
1−t
+
= 1.
≤
n i=1 (1 − t) + tλi (1 − t) + tλi

Thus we have, on the one hand,det(DFt )1/n≥ (1 − t) + tdet(DF1 )

1/n

}1/n
{
voln (B)= (1 − t) + t.
voln (A)(4.3)

On the other hand, the Hölder inequality and the change of variables formula yield
(∫

∫
1/n

det(DFt )
A

dµ0 ≤

)1/n
det(DFt ) dµ0

A

(
=

1
voln (A))1/n∫Ft (A)

d voln.

Therefore we obtain{}1/n {}1/n∫
voln (Ft (A))
voln ((1 − t)A + tB)
1/ndet(DFt ) dµ0 ≤≤.
voln (A)voln (A)A
Combining these, we complete the proof of (4.2).

2

Remark 4.2 We remark that the set (1 − t)A + tB is not necessarily measurable (regardless the measurability of A and B). Hence, to be precise, voln ((1 − t)A + tB) is considered
as an outer measure given by inf W voln (W ), where W ⊂ Rn runs over all measurable setscontaining (1 − t)A + tB. The same remark is applied to Theorems 4.3, 6.1 below.
Next we treat the weighted case (Rn , k · k, m), where m = e−ψ voln with ψ ∈ C ∞ (Rn ).
Then we need to replace 1/n in (4.2) with 1/N for some N ∈ (n, ∞), and the analogueof (4.2) leads us to an important condition on ψ.
Theorem 4.3 (Brunn-Minkowski inequality with weight) Take N ∈ (n, ∞). A
weighted Euclidean space (Rn , k · k, m) with m = e−ψ voln , ψ ∈ C ∞ (Rn ), satisﬁes()1/N≥ (1 − t)m(A)1/N + tm(B)1/N
m (1 − t)A + tB

(4.4)

for all measurable sets A, B ⊂ Rn and all t ∈ [0, 1] if and only ifh∇ψ(x), vi2
Hess ψ(v, v) −≥0
N −nholds for all unit vectors v ∈ Tx Rn .
13

(4.5)

Proof. We ﬁrst prove that (4.5) implies (4.4). Similarly to Theorem 4.1, we assume that
A and B are bounded and of positive measure, and set
µ0 :=

χA
m,
m(A)

µ1 :=

χB
m.
m(B)

We again ﬁnd a convex function f : Rn −→ R such that µt := (Ft )] µ0 is the minimal
geodesic from µ0 to µ1 , where Ft := (1 − t) IdRn +t∇f . Instead of det(DFt ), we consider()()
detm DFt (x) := eψ(x)−ψ(Ft (x)) det DFt (x) .
The coeﬃcient eψ(x)−ψ(Ft (x)) represents the ratio of the weights at x and Ft (x). As in Theorem 4.1 (see, especially, (4.3)), it is suﬃcient to show the concavity of detm (DFt (x))1/N
to derive the desired inequality (4.4). Fix x ∈ A and put
γ(t) := Ft (x),()1/NΦm (t) := detm DFt (x)
,

(
)1/n
Φ(t) := det DFt (x).

On the one hand, it is proved in (4.3) that Φ(t) ≥ (1 − t)Φ(0) + tΦ(1). On the other
hand, the assumption (4.5) implies that e−ψ(Ft (x))/(N −n) is a concave function in t. These
together imply (4.4) via the Hölder inequality. To be precise, we haveΦm (t) = e{ψ(x)−ψ(Ft (x))}/N Φ(t)n/N{}(N −n)/N {}n/N≥ eψ(x)/N (1 − t)e−ψ(x)/(N −n) + te−ψ(F1 (x))/(N −n)(1 − t)Φ(0) + tΦ(1),and then the Hölder inequality yields{}
Φm (t) ≥ eψ(x)/N (1 − t)e−ψ(x)/N Φ(0)n/N + te−ψ(F1 (x))/N Φ(1)n/N= (1 − t)Φm (0) + tΦm (1).
To see the converse, we ﬁx an arbitrary unit vector v ∈ Tx Rn and set γ(t) := x + tv fort ∈ R and a := h∇ψ(x), vi/(N − n). Given ε > 0 and δ ∈ R with ε, |δ| ¿ 1, we consider
two open balls (see Figure 6 where aδ > 0)()()
A+ := B γ(δ), ε(1 − aδ) ,A− := B γ(−δ), ε(1 + aδ) .

B(x, ε)A− = B(γ(−δ), ε(1 + aδ))Figure 6
14

A+ = B(γ(δ), ε(1 − aδ))

Note that A+ = A− = B(x, ε) if δ = 0 and that (1/2)A− + (1/2)A+ = B(x, ε). We
also observe thatm(A± ) = e−ψ(γ(±δ)) cn εn (1 ∓ aδ)n + O(εn+1 ),where cn = voln (B(0, 1)) and O(εn+1 ) is independent of δ. Applying (4.4) to A± witht = 1/2, we obtain
(
)
1
(4.6)
m B(x, ε) ≥ N {m(A− )1/N + m(A+ )1/N }N .
2
We know that m(B(x, ε)) = e−ψ(x) cn εn + O(εn+1 ). In order to estimate the right-handside, we calculate]¯∂ 2 [ −ψ(γ(δ))/N
n/N ¯
e
(1
−
aδ)
¯
∂δ 2{
δ=0
(
) }
Hess ψ(v, v) h∇ψ, vi2h∇ψ, vi n
n n
= −
+
+2
a+
− 1 a2 e−ψ(x)/N
2
N
N
N N
N N
{
}
2
(
)2 e−ψ(x)/Nh∇ψ, vin= − Hess ψ(v, v) +−(N − n)a − h∇ψ, vi.
N −n
N (N − n)N
Due to the choice of a = h∇ψ(x), vi/(N − n) (as the maximizer), we have} −ψ(x)/N{]¯e∂ 2 [ −ψ(γ(δ))/Nh∇ψ(x), vi2
n/N ¯
e
(1
−
aδ)
−
Hess
ψ(v,
v)
.
=
¯
∂δ 2
N −n
N
δ=0
Thus we ﬁnd, by the Taylor expansion of e−ψ(γ(δ))/N (1 − aδ)n/N at δ = 0,m(A− )1/N + m(A+ )1/N
(cn εn )1/N{}h∇ψ, vi2 e−ψ(x)/N 2−ψ(x)/N= 2e− Hess ψ(v, v) −
δ + O(δ 4 ) + O(ε).
N −nNHence we obtain by letting ε go to zero in (4.6) that
[
{
}
]N
1
h∇ψ, vi2 e−ψ(x)/N 2−ψ(x)−ψ(x)/N4e≥ N 2e− Hess ψ(v, v) −
δ + O(δ )
2
N −n
N
[
{
} ]
1
h∇ψ, vi2 2
−ψ(x)
=e
1−
Hess ψ(v, v) −
δ + O(δ 4 ).
2
N −nTherefore we conclude
Hess ψ(v, v) −h∇ψ(x), vi2
≥ 0.
N −n

Applying (4.4) to A = {x}, B = B(x, R) and t = r/R implies( )Nm(B(x, R))R≤
m(B(x, r))
r

2

(4.7)

for all x ∈ Rn and 0 < r < R. Thus, compared with Theorem 3.2, (Rn , k · k, m) satisfying (4.5) behaves like an ‘N -dimensional’ space of nonnegative Ricci curvature (seeTheorem 6.3 for more general theorem in terms of the curvature-dimension condition).
15

4.2

Characterizing lower Ricci curvature bounds

Now we switch to the weighted Riemannian situation (M, g, m), where m = e−ψ volg withψ ∈ C ∞ (M ). Ricci curvature controls volg as we saw in Section 3, and Theorem 4.3
suggests that the quantityh∇ψ, vi2
Hess ψ(v, v) −N −n
has an essential information in controlling the eﬀect of the weight. Their combinationindeed gives the weighted Ricci curvature as follows (cf. [BE], [Qi], [Lo1]).
Deﬁnition 4.4 (Weighted Ricci curvature) Given a unit tangent vector v ∈ Tx M
and N ∈ [n, ∞], the weighted Ricci curvature RicN (v) is deﬁned by{
Ric(v) + Hess ψ(v, v) if h∇ψ(x), vi = 0,
(1) Ricn (v) :=−∞otherwise;(2) RicN (v) := Ric(v) + Hess ψ(v, v) −h∇ψ(x), vi2
for N ∈ (n, ∞);N −n(3) Ric∞ (v) := Ric(v) + Hess ψ(v, v).
We say that RicN ≥ K holds for K ∈ R if RicN (v) ≥ K holds for all unit vectors v ∈ T M .
Note that RicN ≤ RicN 0 holds for n ≤ N ≤ N 0 < ∞. Ric∞ is also called the BakryÉmery tensor. If the weight is trivial in the sense that ψ is constant, then RicN coincides
with Ric for all N ∈ [n, ∞]. One of the most important examples possessing nontrivial
weight is the following.
Example 4.5 (Euclidean spaces with log-concave measures) Consider a weighted
Euclidean space (Rn , k · k, m) with m = e−ψ voln , ψ ∈ C ∞ (Rn ). Then clearly Ric∞ (v) =Hess ψ(v, v), thus Ric∞ ≥ 0 if ψ is convex. The most typical and important example
satisfying Ric∞ ≥ K > 0 is the Gausssian measure( )n/2( )K
2πKn−Kkxk2 /22m=e
voln ,
ψ(x) = kxk + log
.
2π
2
2
K
Note that Hess ψ ≥ K holds independently of the dimension n.
Before stating the main theorem of the section, we mention that optimal transport
in a Riemannian manifold is described in the same manner as the Euclidean spaces (dueto [Mc2], [CMS1]). Given µ0 , µ1 ∈ Pc (M ) with µ0 = ρ0 volg ∈ P ac (M, volg ), there is a(d2 /2)-convex function f : M −→ R such that µt := (Ft )] µ0 with Ft (x) := expx [t∇f (x)],t ∈ [0, 1], gives the unique minimal geodesic from µ0 to µ1 . We do not give the deﬁnition of (d2 /2)-convex functions, but only remark that they are twice diﬀerentiable a.e.
Furthermore, the absolute continuity of µ0 implies that µt is absolutely continuous for
all t ∈ [0, 1), so that we can set µt = ρt volg . Since Ft is diﬀerentiable µ0 -a.e., we canconsider the Jacobian k(DFt )x k (with respect to volg ) which satisﬁes the Monge-Ampèreequation()(4.8)
ρ0 (x) = ρt Ft (x) k(DFt )x k
16

for µ0 -a.e. x.
We next introduce two entropy functionals. Given N ∈ [n, ∞) and an absolutely
continuous probability measure µ = ρm ∈ P ac (M, m), we deﬁne the Rényi entropy as∫
SN (µ) := −
ρ1−1/N dm.
(4.9)
M

We also deﬁne the relative entropy with respect to the reference measure m by∫
Entm (µ) :=ρ log ρ dm.

(4.10)

M

Note that Entm has the opposite sign to the Boltzmann entropy. The domain of thesefunctionals will be extended in the next section ((5.1), (5.2)) to probability measurespossibly with nontrivial singular part. In this section, however, we consider only absolutely
continuous measures for the sake of simplicity. As any two points in Pcac (M, m) areconnected by a unique minimal geodesic contained in Pcac (M, m), the convexity of SN and
Entm in Pcac (M, m) makes sense.
t
Recall (2.3) for the deﬁnition of the function βK,N. The following theorem is due to
von Renesse, Sturm and many others, see Further Reading for more details.
Theorem 4.6 (A characterization of Ricci curvature bound) For a weighted Riemannian manifold (M, g, m) with m = e−ψ volg , ψ ∈ C ∞ (M ), we have RicN ≥ K forsome K ∈ R and N ∈ [n, ∞) if and only if any pair of measures µ0 = ρ0 m, µ1 = ρ1 m ∈Pcac (M, m) satisﬁes
∫
(
)1/N
1−t
SN (µt ) ≤ −(1 − t)βK,Nd(x, y)ρ0 (x)−1/N dπ(x, y)
M ×M
∫
(
)1/N
t
−t
βK,N
d(x, y)ρ1 (y)−1/N dπ(x, y)(4.11)
M ×M

for all t ∈ (0, 1), where (µt )t∈[0,1] ⊂ Pcac (M, m) is the unique minimal geodesic from µ0 toµ1 in the L2 -Wasserstein space (P2 (M ), dW
2 ), and π is the unique optimal coupling of µ0
and µ1 .
Similarly, Ric∞ ≥ K is equivalent to
Entm (µt ) ≤ (1 − t) Entm (µ0 ) + t Entm (µ1 ) −K
2(1 − t)tdW
2 (µ0 , µ1 ) .
2(4.12)

Outline of proof. We give a sketch of the proof for N < ∞ along the lines of [St4] and[LV1]. The case of N = ∞ goes along the essentially same line.
First, we assume RicN ≥ K. Fix µ0 = ρ0 m, µ1 = ρ1 m ∈ Pcac (M, m) and take a(d2 /2)-convex function f : M −→ R such that Ft (x) := expx [t∇f (x)], t ∈ [0, 1], provides
the unique minimal geodesic µt = ρt m = (Ft )] µ0 from µ0 to µ1 . Taking the weight e−ψ
into account, we introduce the Jacobian Jψt (x) := eψ(x)−ψ(Ft (x)) k(DFt )x k with respect tom (like detm in Theorem 4.3). Then it follows from the Monge-Ampère equation (4.8)
with respect to volg that()
(4.13)
ρ0 (x) = ρt Ft (x) Jψt (x)17for µ0 -a.e. x (i.e., the Monge-Ampère equation with respect to m).
Now, the essential point is that optimal transport is performed along geodesics t 7−→
expx [t∇f (x)] = Ft (x). Therefore its variational vector ﬁelds are Jacobi ﬁelds (recall (3.1)),
and controlled by Ricci curvature. Together with the weight control as in Theorem 4.3,
calculations somewhat similar to (but more involved than) Theorem 3.2 shows our key
inequality()1/N)1/N ψ 1/N(
1−ttJψt (x)1/N ≥ (1 − t)βK,Nd(x, F1 (x))+ tβK,Nd(x, F1 (x))
J1 (x) .

(4.14)

This inequality can be thought of as an inﬁnitesimal version of the Brunn-Minkowski
inequality (see (4.4) and Theorem 6.1(i) as well). As the change of variables formula andthe Monge-Ampère equation (4.13) yield∫
SN (µt ) = −ρt (Ft )1−1/NJψt dm = −M∫ ( ψ )1/N
Jt
dµ0 ,
ρ0
M

we obtain from (4.14) (and (4.13) again) that
∫

1−t
βK,N
(d(x, F1 (x)))1/N
SN (µt ) ≤ −(1 − t)dµ0 (x)ρ0 (x)1/N
M
∫
t
βK,N
(d(x, F1 (x)))1/N
dµ0 (x).
−t
ρ1 (F1 (x))1/N
M

This is the desired inequality (4.11), for π = (IdM ×F1 )] µ0 .
Second, we assume (4.11). Then applying it to uniform distributions on balls (asin the proof of Theorem 4.3) shows RicN ≥ K. More precisely, we use the generalized
Brunn-Minkowski inequality (Theorem 6.1) instead of (4.4).
2t
As β0,N≡ 1, the inequality (4.11) is simpliﬁed into the convexity of SN

SN (µt ) ≤ (1 − t)SN (µ0 ) + tSN (µ1 )when K = 0. For K 6= 0, however, the K-convexity of SN
SN (µt ) ≤ (1 − t)SN (µ0 ) + tSN (µ1 ) −K
2(1 − t)tdW
2 (µ0 , µ1 )2turns out uninteresting (see [St2, Theorem 1.3]). This is a reason why we need to consider
a more subtle inequality like (4.11).
Theorem 4.6 gives an answer to Question 3.5, for the conditions (4.11), (4.12) are
written in terms of only distance and measure, without using the diﬀerentiable structure.
Then it is interesting to consider these conditions for general metric measure spaces as‘synthetic Ricci curvature bounds’, and we should verify the stability. We discuss themin the next section.
Further Reading We refer to [AGS], [Vi1] and [Vi2, Part I] for the basics of optimaltransport theory and Wasserstein geometry. McCann’s [Mc2] fundamental result on the18

shape of optimal transport maps is generalized to not necessarily compactly supported
measures in [FF] and [FG] (see also [Vi2, Chapter 10]).
See [Ga] and [Le, Section 2.2] for the Brunn-Minkowski inequality and related topics.
The Bakry-Émery tensor Ric∞ was introduced in [BE], and its generalization RicN is due
to Qian [Qi]. See also [Lo1] for geometric and topological applications, [Mo, Chapter 18]
and the references therein for minimal surface theory in weighted manifolds (which arecalled manifolds with density there).
After McCann’s [Mc1] pinoneering work on the convexity of the relative entropy along
geodesics in the Wasserstein space (called the displacement convexity) over Euclidean
spaces, Cordero-Erausquin, McCann and Schmuckenschläger [CMS1] ﬁrst showed that
Ric ≥ 0 implies (4.12) with K = 0 in unweighted Riemannian manifolds. They [CMS2]
further proved that Ric∞ ≥ K implies (4.12) in the weighted situation. Then Theorem 4.6 is due to von Renesse and Sturm [vRS], [St2] for N = ∞, and independently to
Sturm [St3], [St4] and Lott and Villani [LV2], [LV1] for N < ∞.
We comment on recent work on a variant of (4.11). Studied in [BaS1] is the following
inequality (called the reduced curvature-dimension condition)
∫
(
)1/N
1−t
SN (µt ) ≤ −(1 − t)βK,Nρ0 (x)−1/N dπ(x, y)+1 d(x, y)
M ×M
∫
(
)1/N
t
−t
βK,N
ρ1 (y)−1/N dπ(x, y).
(4.15)+1 d(x, y)
M ×M

Note the diﬀerence betweenttβK,N(r)1/N = t1/N(

sK,N (tr)sK,N (r))1−1/N
,

t
1/N
tβK,N
=
+1 (r)

sK,N +1 (tr).
sK,N +1 (r)We remark that (4.15) coincides with (4.11) when K = 0, and is weaker than (4.11) for
general K 6= 0. The condition (4.15) is also equivalent to RicN ≥ K for Riemannian manifolds. In the setting of metric measure spaces, (4.15) has some advantages such as thetensorization and the localization properties (see Subsection 8.3 for more details). One
drawback is that, as it is weaker than (4.11), (4.15) derives slightly worse estimates than
(4.11) (in the Bishop-Gromov volume comparison (Theorem 6.3), the Bonnet-Myers diameter bound (Theorem 6.5), the Lichnerowicz inequality (Theorem 6.7) etc.). Nevertheless,such weaker estimates are suﬃcient for several topological applications.
See also [St2] and [OT] for the K-convexity of generalized entropies (or free energies)
and its characterization and applications. It is discussed in [St2, Theorem 1.7] that there
is a class of functionals whose K-convexity is equivalent to Ric ≥ K and dim ≤ N for
unweighted Riemannian manifolds. The choice of a functional is by no means unique, and
it is unclear how this observation relates to the curvature-dimension condition.

5

The curvature-dimension condition and stability

Motivated by Theorem 4.6, we introduce the curvature-dimension condition for metricmeasure spaces and show that it is stable under the measured Gromov-Hausdorﬀ convergence. In this and the next sections, (X, d, m) will always be a metric measure space inthe sense of Section 2.
19

5.1

The curvature-dimension condition

We can regard the conditions (4.11), (4.12) as convexity estimates of the functionals SN
and Entm . For the sake of consistency with the monotonicity of RicN in N (RicN ≤ RicN 0
for N ≤ N 0 ), we introduce important classes of functionals (due to McCann [Mc1])
including SN and Entm .
For N ∈ [1, ∞), denote by DC N (displacement convexity class) the set of continuous
convex functions U : [0, ∞) −→ R such that U (0) = 0 and that the function ϕ(s) =sN U (s−N ) is convex on (0, ∞). Similarly, deﬁne DC ∞ as the set of continuous convex
functions U : [0, ∞) −→ R such that U (0) = 0 and that ϕ(s) = es U (e−s ) is convex
on R. In both cases, the convexity of U implies that ϕ is non-increasing. Observe themonotonicity, DC N 0 ⊂ DC N holds for 1 ≤ N ≤ N 0 ≤ ∞.
For µ ∈ P(X), using its Lebesgue decomposition µ = ρm + µs into absolutely continuous and singular parts, we set∫
U (r)Um (µ) :=U (ρ) dm + U 0 (∞)µs (X),U 0 (∞) := lim
.
r→∞
r
X
Note that U 0 (∞) indeed exists as U (r)/r is non-decreasing. In the case where U 0 (∞) = ∞,we set ∞ · 0 := 0 by convention. The most important element of DC N is the function
U (r) = N r(1 − r−1/N ) which induces the Rényi entropy (4.9) in a slightly deformed form
as
(
)
∫
∫
−1/N
s
1−1/N
Um (µ) = N
ρ(1 − ρ) dm + N µ (X) = N 1 −
ρ
dm .
(5.1)
X

X

This extends (4.9) to whole P(X). Letting N go to inﬁnity, we have U (r) = r log r ∈ DC ∞as well as the relative entropy (extending (4.10))∫
Um (µ) =ρ log ρ dm + ∞ · µs (X).
(5.2)
X

Let us denote by Γ(X) the set of minimal geodesics γ : [0, 1] −→ X endowed with the
distance()dΓ(X) (γ1 , γ2 ) := sup dX γ1 (t), γ2 (t) .
t∈[0,1]Deﬁne the evaluation map et : Γ(X) −→ X for t ∈ [0, 1] as et (γ) := γ(t), and note that
this is 1-Lipschitz. A probability measure Π ∈ P(Γ(X)) is called a dynamical optimaltransference plan if π := (e0 × e1 )] Π is an optimal coupling of α(0) and α(1). Thenthe curve α(t) := (et )] Π, t ∈ [0, 1], is a minimal geodesic in (P2 (X), dW
2 ). We remarkthat Π is not uniquely determined by α and π, that is to say, diﬀerent plans Π and Π0
could generate the same minimal geodesic α and optimal coupling π. If (X, d) is locallycompact (and hence proper), then any minimal geodesic in P2 (X) is associated with a(not necessarily unique) dynamical optimal transference plan ([LV2, Proposition 2.10],[Vi2, Corollary 7.22]).
Now we are ready to present the precise deﬁnition of the curvature-dimension conditionin the form due to Lott and Villani (after Sturm and others, see Further Reading of thisand the previous sections).
20

Deﬁnition 5.1 (The curvature-dimension condition) Suppose that m(B(x, r)) ∈(0, ∞) holds for all x ∈ X and r ∈ (0, ∞). For K ∈ R and N ∈ (1, ∞], we say thata metric measure space (X, d, m) satisﬁes the curvature-dimension condition CD(K, N )if, for any µ0 = ρ0 m + µs0 , µ1 = ρ1 m + µs1 ∈ Pb (X), there exists a dynamical optimal transference plan Π ∈ P(Γ(X)) associated with a minimal geodesic α(t) = (et )] Π, t ∈ [0, 1],from µ0 to µ1 and an optimal coupling π = (e0 × e1 )] Π of µ0 and µ1 such that we have
)
(
∫
(
)
(
)
ρ0 (x)1−tdπx (y)dm(x)Um α(t) ≤ (1 − t)βK,N d(x, y) U
1−tβK,N(d(x, y))
X×X
(
)
∫
(
)
ρ1 (y)t+t
βK,N d(x, y) U
dπy (x)dm(y)tβK,N(d(x, y))X×X
+ U 0 (∞){(1 − t)µs0 (X) + tµs1 (X)}

(5.3)

for all U ∈ DC N and t ∈ (0, 1), where πx and πy denote disintegrations of π by µ0 and µ1 ,i.e., dπ(x, y) = dπx (y)dµ0 (x) = dπy (x)dµ1 (y).
tIn the special case of K = 0, as β0,N≡ 1, the inequality (5.3) means the convexity ofUm()
Um α(t) ≤ (1 − t)Um (µ0 ) + tUm (µ1 ),without referring to the optimal coupling π. In the case where both µ0 and µ1 are
absolutely continuous, we have dπ(x, y) = ρ0 (x)dπx (y)dm(x) = ρ1 (y)dπy (x)dm(y) andhence (5.3) is rewritten in a more symmetric form as
)
(
1−t
βK,N
(d(x, y))ρ0 (x)
Um α(t) ≤ (1 − t)dπ(x, y)U1−tρ0 (x)βK,N(d(x, y))
X×X
(
)
∫
t
βK,N
(d(x, y))ρ1 (y)+tUdπ(x, y).
t
ρ1 (y)βK,N(d(x, y))
X×X
(

)

∫

(5.4)

Note that choosing U (r) = N r(1 − r−1/N ) and U (r) = r log r in (5.4) reduce to (4.11) and(4.12), respectively. We summarize remark on and the background of Deﬁnition 5.1.
Remark 5.2 (a) It is easily checked that, if (X, d, m) satisﬁes CD(K, N ), then the scaled
metric measure space (X, cd, c0 m) for c, c0 > 0 satisﬁes CD(K/c2 , N ).
(b) In Deﬁnition 5.1, to be precise, we need to impose the condition√()
m X \ B(x, π (N − 1)/K) = 0t
for all x ∈ X if K > 0 and N < ∞, in order to stay inside the domain of βK,N.
Nevertheless, this is always the case by virtue of the generalized Bonnet-Myers theorem(Theorem 6.5) below.
t(c) Recall that U (r)/r is non-decreasing, and observe that βK,N(r) is increasing in K
and decreasing in N . Combining this with the monotonicity DC N 0 ⊂ DC N for N ≤ N 0(and (b) above), we see that CD(K, N ) implies CD(K 0 , N 0 ) for all K 0 ≤ K and N 0 ≥ N .
Therefore, in the condition CD(K, N ), K represents a lower bound of the Ricci curvature
and N represents an upper bound of the dimension.

21

(d) The validity of (5.3) along only ‘some’ geodesic is essential to establish the stability.
In fact, if we impose it for all geodesics between µ0 and µ1 , then it is not stable underconvergence (in the sense of Theorem 5.6). This is because, when a sequence {(Xi , di )}i∈N
converges to the limit space (X, d), there may be a geodesic in X which can not be
represented as the limit of a sequence of geodesics in Xi . Therefore the convexity along
geodesics in Xi does not necessarily imply the convexity along all geodesics in X. One
typical example is a sequence of `np -spaces as p goes to 1 or ∞. Only straight lines are
geodesics in `np with 1 < p < ∞, while `n1 and `n∞ have much more geodesics. (See Figure 7,
where γi for all i = 0, . . . , 3 are geodesic for `21 , while only the straight line segment γ0 is
geodesic for `2p with 1 < p < ∞.)
6

γ1

x
γ2
γ0
γ3
-

y
Figure 7(e) In Riemannian manifolds or, more generally, non-branching proper metric measurespaces, we can reduce (5.3) to a special case from two aspects as follows. If (5.4) holds
for U (r) = N r(1 − r−1/N ) (or U (r) = r log r if N = ∞) and all measures in Pbac (X, m)(and hence in Pcac (X, m)) with continuous densities, then (5.3) holds for all U ∈ DC N andall measures in Pb (X) ([St4, Proposition 4.2], [LV2, Proposition 3.21, Lemma 3.24]). In
this sense, (4.11) and (4.12) are essential among the class of inequalities (5.3). A geodesic
space is said to be non-branching if geodesics in it do not branch (see Subsection 8.1 forthe precise deﬁnition).
(f) In Riemannian manifolds, (4.14) implies (5.4) for all U ∈ DC N . Indeed, α(t) = ρt m
is absolutely continuous and the change of variables formula and the Monge-Ampèreequation (4.13) imply
( ) ψ
∫
∫
∫
(
)
(
) ψ
ρ0 J tdµ0 .
Um α(t) =U (ρt ) dm =U ρt (Ft ) Jt dm =UJψt ρ0
M
M
M
For N < ∞, as ϕ(s) = sN U (s−N ) is non-increasing and convex, (4.14) yields
(
)
∫
1−t
t
(
)
βK,N
(d(x, F1 (x)))1/N
βK,N(d(x, F1 (x)))1/N
Um α(t) ≤ϕ (1 − t)+tdµ0 (x)ρ0 (x)1/N
ρ1 (F1 (x))1/N
M
)
( t
)}
( 1−t
∫ {
βK,N (d(x, F1 (x)))1/N
βK,N (d(x, F1 (x)))1/N+ tϕ
dµ0 (x).
≤(1 − t)ϕρ0 (x)1/N
ρ1 (F1 (x))1/N
M
22

The case of N = ∞ is similar. This means that the inﬁnitesimal expression of CD(K, N )is always (4.14) whatever U is, and various ways of integration give rise to the deﬁnitionof CD(K, N ) involving DC N .
Although Jψt relies on the diﬀerentiable structure of M , it is possible to rewrite (4.14)
through (4.13) as
(
)−1/N
(
)1/N
1−t
ρt Ft (x)≥ (1 − t)βK,Nd(x, F1 (x))
ρ0 (x)−1/N()1/N ()−1/Nt+ tβK,Nd(x, F1 (x))
ρ1 F1 (x)(see [St4, Proposition 4.2(iv)]). This makes sense in metric measure spaces, however, theintegrated inequalities (i.e., (5.3), (5.4)) are more convenient for verifying the stability.
(g) The role of the dynamical optimal transference plan Π may seem unclear in Deﬁnition 5.1, as only α and π appear in (5.3). We use only α and π also in applications in
Section 6. As we shall see in Theorem 5.6, it is the stability in which Π plays a crucial
role.

5.2

Stability and geometric backgroundAs we mentioned in Remark 3.1, one geometric motivation behind the curvature-dimension
condition is the theory of Alexandrov spaces. That is to say, we would like to ﬁnd a wayof formulating and investigating singular spaces of Ricci curvature bounded below insome sense (recall Question 3.5). Then, what kind of singular spaces should we consider?
Here comes into play another deep theory of the precompactness with respect to theconvergence of metric (measure) spaces. Brieﬂy speaking, the precompactness ensuresthat a sequence of Riemannian manifolds with a uniform lower Ricci curvature boundcontains a convergent subsequence. Such a limit space is not a manifold any more, butshould inherits some properties. In order to make use of the curvature-dimension conditionin the limit, we need to establish that it is preserved under the convergence (actually, thelimit of Alexandrov spaces is again an Alexandrov space.)
We say that a map ϕ : Y −→ X between metric spaces is ε-approximating for ε > 0 if
¯ (¯)
¯dX ϕ(y), ϕ(z) − dY (y, z)¯ ≤ εholds for all y, z ∈ Y and if B(ϕ(Y ), ε) = X.
Deﬁnition 5.3 (Measured Gromov-Hausdorﬀ convergence) Consider a sequenceof metric measure spaces {(Xi , di , mi )}i∈N and another metric measure space (X, d, m).
(1) (Compact case) Assume that (Xi , di ) for all i ∈ N and (X, d) are compact. We saythat {(Xi , di , mi )}i∈N converges to (X, d, m) in the sense of the measured GromovHausdorﬀ convergence if there are sequences of positive numbers {εi }i∈N and Borel
maps {ϕi : Xi −→ X}i∈N such that limi→∞ εi = 0, ϕi is an εi -approximating map,and that (ϕi )] mi weakly converges to m.
(2) (Noncompact case) Assume that (Xi , di ) for all i ∈ N and (X, d) are proper, andﬁx base points xi ∈ Xi and x ∈ X. We say that {(Xi , di , mi , xi )}i∈N converges to23

(X, d, m, x) in the sense of the pointed measured Gromov-Hausdorﬀ convergence if,
for all R > 0, {(B(xi , R), di , mi )}i∈N converges to (B(x, R), d, m) in the sense of themeasured Gromov-Hausdorﬀ convergence (as in (1) above).
If we consider only distance structures (Xi , di ) and (X, d) and remove the weak convergence condition on ϕi , then it is the (pointed) Gromov-Hausdorﬀ convergence under which
the lower sectional curvature bound in the sense of Alexandrov is known to be preserved.
The following observation ([LV2, Proposition 4.1]) says that the Gromov-Hausdorﬀ convergence of a sequence of metric spaces is propagated to the Wasserstein spaces overthem.
Proposition 5.4 If a sequence of compact metric spaces {(Xi , di )}i∈N converges to acompact metric space (X, d) in the sense of the Gromov-Hausdorﬀ convergence, then soWdoes the sequence of Wasserstein spaces {(P(Xi ), dW2 )}i∈N to (P(X), d2 ).
More precisely, εi -approximating maps ϕi : Xi −→ X give rise to θ(εi )-approximating
maps (ϕi )] : P(Xi ) −→ P(X) such that θ is a universal function satisfying limε↓0 θ(ε) = 0.
In the noncompact case, the pointed Gromov-Hausdorﬀ convergence of {(Xi , di , xi )}i∈N
to (X, d, x) similarly implies the Gromov-Hausdorﬀ convergence of {(P(B(xi , R)), dW2 )}i∈NW
to (P(B(x, R)), d2 ) for all R > 0 (instead of the convergence of (B(δxi , R), dW
2 ) to(B(δx , R), dW)).
2
The following inspiring precompactness theorem is established by Gromov [Gr, Section 5.A] for the Gromov-Hausdorﬀ convergence, and extended by Fukaya [Fu] to themeasured case.
Theorem 5.5 (Gromov-Fukaya precompactness) Let {(Mi , gi , volgi , xi )}i∈N be a sequence of pointed Riemannian manifolds such thatRicgi ≥ K,

dim Mi ≤ N

uniformly hold for some K ∈ R and N ∈ N. Then it contains a subsequence which isconvergent to some pointed proper metric measure space (X, d, m, x) in the sense of thepointed measured Gromov-Hausdorﬀ convergence.
To be more precise, we choose a complete space as the limit, and then the properness
follows from our hypotheses Ric ≥ K and dim ≤ N . The key ingredient of the proof is the
Bishop-Gromov volume comparison (Theorem 3.2) from which we derive an upper boundof the doubling constant supx∈M, r≤R volg (B(x, 2r))/ volg (B(x, r)) for each R ∈ (0, ∞).
By virtue of Theorem 5.5, starting from a sequence of Riemannian manifolds with auniform lower Ricci curvature bound, we ﬁnd the limit space of some subsequence. Sucha limit space is not a manifold any more, but should have some inherited properties.
The stability of the curvature-dimension condition ensures that we can use it for theinvestigation of these limit spaces.
We remark that the measures of balls volgi (B(xi , R)) for i ∈ N have a uniform upperbound depending only on K, N and R thanks to Theorem 3.2. However, it could tend
to zero, and then we can not obtain any information on (X, d, m). Therefore we shouldtake a scaling ci volgi with some appropriate constant ci > 1. It does not change anything
24because the weighted Ricci curvature is invariant under scalings of the measure (the weightfunction of m̃ = cm is ψm̃ = ψm − log c). By the same reasoning, it is natural to assume
that any bounded open ball has a ﬁnite positive measure in the next theorem (see also
Remark 5.2(a)).
Theorem 5.6 (Stability) Assume that a sequence of pointed proper metric measurespaces {(Xi , di , mi , xi )}i∈N uniformly satisﬁes CD(K, N ) for some K ∈ R and N ∈ (1, ∞]and that it converges to a pointed proper metric measure space (X, d, m, x) in the sense of
the pointed measured Gromov-Hausdorﬀ convergence. If, moreover, 0 < m(B(x, r)) < ∞holds for all x ∈ X and r ∈ (0, ∞), then (X, d, m) satisﬁes CD(K, N ).
Outline of proof. The proof of stability goes as follows (along the lines of [LV2], [LV1]).
First of all, as we consider only measures with bounded (and hence compact) support inDeﬁnition 5.1, we can restrict ourselves to measures with continuous density and compact
support. Indeed, it implies by approximation the general case ([LV2, Proposition 3.21,
Lemma 3.24], see also Remark 5.2(e)). Given continuous measures µ = ρm, ν = σm ∈Pcac (X, m) and εi -approximating maps ϕi : Xi −→ X as in Deﬁnition 5.3, we considerµi = ∫ρ ◦ ϕi· mi ,
ρ
◦
ϕ
dm
i
i
Xi

νi = ∫σ ◦ ϕi· mi ∈ Pcac (Xi , mi )σ ◦ ϕi dmi
Xi

and take a dynamical optimal transference plan Πi ∈ P(Γ(Xi )) from µi to νi satisfying(5.4). Note that (ϕi )] µi and (ϕi )] νi weakly converge to µ and ν, respectively, thanks tothe continuity of ρ and σ.
By a compactness argument ([LV1, Theorem A.45]), extracting a subsequence if necessary, Πi converges to some dynamical transference plan Π ∈ P(Γ(X)) from µ to ν suchthat, setting
αi (t) := (et )] Πi ,α(t) := (et )] Π,

πi := (e0 × e1 )] Πi ,π := (e0 × e1 )] Π,(ϕi )] αi and (ϕi × ϕi )] πi weakly converge to α and π, respectively. Then it follows from
Proposition 5.4 that α is a minimal geodesic from µ to ν and that π is an optimal couplingof µ and ν.
On the one hand, the right-hand side of (5.4) for πi converges to that for π by virtueof the continuous densities. On the other hand, the monotonicity)()(
U(ϕi )] mi (ϕi )] [αi (t)] ≤ Umi αi (t)and the lower semi-continuity()
Um (α(t)) ≤ lim inf U(ϕi )] mi (ϕi )] [αi (t)]i→∞hold true in general ([LV2, Theorem B.33]). Therefore we obtain (5.4) for Π and complete
the proof.
2
Further Reading The deﬁnition of the curvature-dimension condition is much indebted
to McCann’s inﬂuential work [Mc1] introducing the important class of functions DC N as25

well as the displacement convexity along geodesics in the Wasserstein space (see Further Reading in Section 4). Otto and Villani’s work [OV] on the relation between such
convexity of the entropy and several functional inequalities was also inspiring.
The term ‘curvature-dimension condition’ is used by Sturm [St3], [St4] (and also in[Vi2]) following Bakry and Émery’s celebrated work [BE]. Sturm’s condition requires that
(5.3) is satisﬁed for all absolutely continuous measures and U = SN 0 for all N 0 ∈ [N, ∞].
Lott and Villani [LV2], [LV1], independently of Sturm, introduced the condition as in
Theorem 4.6 and call it N -Ricci curvature bounded from below by K. These conditions
are equivalent in non-branching spaces (see Remark 5.2(e) and Subsection 8.1). In locally compact non-branching spaces, it is also possible to extend (5.3) from compactly
supported measures to not necessarily compactly supported measures (see [FV]).
See [Fu], [Gr, Chapter 3, Section 5.A] and [BBI, Chapters 7, 8] for the basics of (measured) Gromov-Hausdorﬀ convergence and for precompactness theorems. The stability
under the measured Gromov-Hausdorﬀ convergence we presented above is due to Lottand Villani [LV2], [LV1]. Sturm [St3], [St4] also proved the stability with respect to a different, his own notion of convegence induced from his D-distance between metric measurespaces. Roughly speaking, the D-distance takes couplings not only for measures, but also
for distances (see [St3] for more details).
We also refer to celebrated work of Cheeger and Colding [CC] (mentioned in Further Reading of Section 3) for related geometric approach toward the investigation oflimit spaces of Riemannian manifolds of Ricci curvature bounded below. Their strategy
is to fully use the fact that it is the limit of Riemannian manifolds. They reveal thedetailed local structure of such limit spaces, however, it also turns out that the limit
spaces can have highly wild structures (see a survey [We] and the references therein). We
can not directly extend Cheeger and Colding’s theory to metric measure spaces with thecurvature-dimension condition. Their key tool is the Cheeger-Gromoll type splitting theorem, but Banach spaces prevent us to apply it under the curvature-dimension condition(see Subsection 6.3(C) for more details).

6

Geometric applications

Metric measure spaces satisfying the curvature-dimension condition CD(K, N ) enjoy many
properties common to ‘spaces of dimension ≤ N and Ricci curvature ≥ K’. To be more
precise, though N ∈ (1, ∞] is not necessarily an integer, we will obtain estimates numerically extended to non-integer N . Proofs based on optimal transport theory themselves
are interesting and inspiring. Although we concentrate on geometric applications in thisarticle, there are also many analytic applications including the Talagrand inequality, logarithmic Sobolev inequality (and hence the normal concentration of measures), global
Poincaré inequality and so forth (see [LV2], [LV1]).

6.1

Generalized Brunn-Minkowski inequality and applications

Our ﬁrst application is a generalization of the Brunn-Minkowski inequality (4.2), (4.4) tocurved spaces. This follows from the curvature-dimension condition (5.4) applied to SN
26and Entm (i.e., (4.11) and (4.12)) between uniform distributions on two measurable sets.
In the particular case of K = 0, we obtain the concavity of m1/N or log m as in (4.2),(4.4). Given two sets A, B ⊂ X and t ∈ (0, 1), we denote by Zt (A, B) the set of points
γ(t) such that γ : [0, 1] −→ X is a minimal geodesic with γ(0) ∈ A and γ(1) ∈ B. We
remark that Zt (A, B) is not necessarily measurable regardless the measurability of A andB, however, it is not a problem because m is regular (see Remark 4.2).
The following theorem is essentially contained in von Renesse and Sturm [vRS] forN = ∞, and due to Sturm [St4] for N < ∞. Again
we will be implicitly indebted√
to Theorem 6.5 that guarantees that diam X ≤ π (N − 1)/K if K > 0 and N < ∞(see Remark 5.2(b)). Figure 8 represents rough image of the theorem, Z1/2 (A, B) has
more measure in a positively curved space, and less measure in a negatively curved space(compare this with Figure 1).

A

Z1/2 (A, B)

B

A

B

Z1/2 (A, B)
Ric > 0

Ric < 0Figure 8

Theorem 6.1 (Generalized Brunn-Minkowski inequality) Take a metric measurespace (X, d, m) satisfying CD(K, N ) and two measurable sets A, B ⊂ X.
(i) If N ∈ (1, ∞), then we have(
)1/N
m Zt (A, B)≥ (1 − t)
+t

(
)1/N
1−t
βK,N
d(x, y)· m(A)1/N
x∈A, y∈B()1/Ntinf βK,Nd(x, y)· m(B)1/N
inf

x∈A, y∈B

for all t ∈ (0, 1).
(ii) If N = ∞ and 0 < m(A), m(B) < ∞, then we have()
log m Zt (A, B)

(
)2
χA
χB
K
W
m,
m
≥ (1 − t) log m(A) + t log m(B) + (1 − t)td2
2m(A) m(B)for all t ∈ (0, 1).
27

Proof. Similarly to Theorem 4.1, we can assume that A and B are bounded and of
positive measure. Set
µ0 :=

χA
· m,
m(A)

µ1 :=

χB
· m,
m(B)

β̂ t :=

infx∈A, y∈B

(
)
t
βK,N
d(x, y) .

(i) We consider U (r) = N r(1 − r−1/N ) and recall from (5.1) that, for µ = ρm + µs ,
(
)
∫
1−1/N
Um (µ) = N 1 −
ρ
dm .
X

Hence it follows from CD(K, N ) that there is a minimal geodesic α : [0, 1] −→ P(X) fromµ0 to µ1 as well as an optimal coupling π such that, for all t ∈ (0, 1),
∫
∫
{
(
)}1/N
1−1/N
1−t
ρt
dm ≤ −(1 − t)m(A)βK,N−d(x, y)dπ(x, y)
X
X×X
∫
{
(
)}1/N
t
−t
m(B)βK,Nd(x, y)dπ(x, y)X×X≤ −(1 − t)(β̂ 1−t )1/N m(A)1/N − t(β̂ t )1/N m(B)1/N ,where we set α(t) = ρt m + µst . Then the Hölder inequality yields
∫

−1/N
ρt
· ρt dm ≤

(∫

)1/N
ρ−1
t · ρt dm()1/N= m(supp ρt )1/N ≤ m Zt (A, B).

supp ρt

X

This completes the proof for N < ∞.
(ii) We argue similarly and obtain from (4.12) that()
K
2
Entm α(t) ≤ −(1 − t) log m(A) − t log m(B) − (1 − t)tdW
2 (µ0 , µ1 ) .
2Note that, since Entm (α(t)) < ∞, α(t) is absolutely continuous and written as α(t) = ρt m.
Furthermore, Jensen’s inequality applied to the convex function s 7−→ s log s shows∫()
dm
Entm α(t) = m(supp ρt )
ρt log ρtm(supp ρt )supp ρt
(∫
)
∫
dm
dm
≥ m(supp ρt )
ρt
· log
ρt
m(supp ρt )m(supp ρt )supp ρtsupp ρt()= − log m(supp ρt ) ≥ − log m Zt (A, B) .
2

We complete the proof.
As a corollary, we ﬁnd that m has no atom unless X consists of a single point.

Corollary 6.2 If (X, d, m) contains more than two points and if it satisﬁes CD(K, N )with some K ∈ R and N ∈ (1, ∞], then any one point set {x} ⊂ X has null measure.

28

Proof. It is suﬃcient to show the case of N = ∞ (due to Remark 5.2(c)). Put A = {x}and assume m({x}) > 1 (due to Remark 5.2(a)). Take r > 0 with m(B(x, 2r)\B(x, r)) > 0(it is the case for small r > 0), and note that)(
Zt {x}, B(x, 2r) \ B(x, r) ⊂ B(x, 2tr) \ B(x, tr)for all t ∈ (0, 1). Thus we apply Theorem 6.1(ii) with t = 2−k , k ∈ N, and ﬁnd()
log m B(x, 21−k r) \ B(x, 2−k r)
(
)
(
) |K|
≥ (1 − 2−k ) log m({x}) + 2−k log m(B(x, 2r) \ B(x, r)) −(1 − 2−k )2−k (2r)2 .
2Summing this up in k ∈ N, we observe∞∑()
log m B(x, 21−k r) \ B(x, 2−k r) = ∞.

k=1

This is a contradiction since we have
∞
∑
k=1

∞
(
)
(
) ∑
m B(x, 21−k r) \ B(x, 2−k r)log m B(x, 21−k r) \ B(x, 2−k r) ≤()= m B(x, 2r) \ {x} < ∞.
k=1

2
Under CD(K, N ) of the ﬁnite dimension N < ∞, applying Theorem 6.1(i) to thinannuli shows a generalization of the Bishop-Gromov volume comparison theorem (Theorem 3.2, see also (4.7)).
Theorem 6.3 (Generalized Bishop-Gromov volume comparison) Suppose that ametric measure space (X, d, m) satisﬁes CD(K, N ) with K ∈ R and N ∈ (1, ∞). Then
we have∫R
sK,N (t)N −1 dtm(B(x, R))0∫≤ rm(B(x, r))s (t)N −1 dt
0 K,N√
for all x ∈ X and 0 < r < R (≤ π (N − 1)/K if K > 0).
Proof. The proof is essentially the same as the Riemannian case. We apply Theorem 6.1(i) to concentric thin annuli and obtain an estimate corresponding to the Bishop
area comparison of concentric spheres (3.4). Then we take the sum and the limit (insteadof integration), and obtain the theorem.
We give more detailed calculation for thoroughness. For any annulus B(x, r2 )\B(x, r1 )and t ∈ (0, 1), Theorem 6.1(i) and Corollary 6.2 yield that
)N −1
(
(
)
(
)
sK,N (td)Nm B(x, r2 ) \ B(x, r1 )m B(x, tr2 ) \ B(x, tr1 ) ≥ t
infd∈[r1 ,r2 ]tsK,N (d))
inf d∈[r1 ,r2 ] sK,N (td)N −1 (
≥t·
m
B(x,
r
)
\
B(x,
r
)
.
(6.1)
2
1
supd∈[r1 ,r2 ] sK,N (d)N −1
29

This corresponds to (3.4) in the Riemannian case. Set h(t) := sK,N (t)N −1 for brevity, andput tL := (r/R)1/L < 1 for L ∈ N. Applying (6.1) to r1 = tL r, r2 = r and t = tl−1forLl ∈ N, we have
∞
) ∑
(
)
l
m B(x, r) =m B(x, tl−1
r)
\
B(x,
t
r)
L
L

(

l=1

≥

{∑
∞

}
l−1
(
)
l−1 inf d∈[tL r,r] h(tL d)tLm B(x, r) \ B(x, tL r) .
supd∈[tL r,r] h(d)
l=1

l−1−L
We similarly deduce from (6.1) with r1 = tl−Lr and t = tLL−l+1 for l =L r, r2 = tL
1, . . . , L thatL)∑m B(x, r) \ B(x, tL r)
tl−1
L

(

l=1

≥ tLL

inf

h(d)

d∈[tL r,r]

L
∑

sup

h(d)

l−L
l−1−L
d∈[tL
r,tL
r]

(
)
m B(x, tl−1−Lr) \ B(x, tl−L
L
L r)

l=1

(
)
r
=
inf h(d) · m B(x, R) \ B(x, r) .
R d∈[tL r,r]Combining these, we obtain
L
) ∑
l
(tl−1
m B(x, r) ·L − tL )R

(

l=1

(
)
= (1 − tL )R · m B(x, r) ·≥ (1 − tL )R ·

{∑
∞

sup
d∈[tL R,R]

L
∑
l=1

tl−1
L

h(tl−1
L d)
sup

h(d)

l−1−L
d∈[tl−Lr]
L r,tL}l−1()
rl−1 inf d∈[tL r,r] h(tL d)
tL·
inf h(d) · m B(x, R) \ B(x, r)
supd∈[tL r,r] h(d)R d∈[tL r,r]
l=1∞
) inf d∈[tL r,r] h(d) ∑(tl−1 − tlL )r inf h(tl−1≥ m B(x, R) \ B(x, r) ·L d).
d∈[tL r,r]
supd∈[tL r,r] h(d) l=1 L(

Letting L diverge to inﬁnity shows
∫
∫
(
) R
(
) r
N −1
m B(x, r)sK,N (t)dt ≥ m B(x, R) \ B(x, r)
sK,N (t)N −1 dt.
r

(6.2)

0

This corresponds to (3.5) in the Riemannian case, and the same calculation as the last
step of the proof of Theorem 3.2 completes the proof.
2
Theorem 6.3 shows that the doubling constant supx∈X, r≤R m(B(x, 2r))/m(B(x, r)) isbounded for each R ∈ (0, ∞), therefore X is proper (see also the paragraph following
Theorem 5.5).
Corollary 6.4 Assume that (X, d, m) satisﬁes CD(K, N ) for some K ∈ R and N ∈(1, ∞). Then (X, d) is proper.
30

Next we generalize the Bonnet-Myers diameter bound (Corollary 3.4). We remarkthat the proof√ below uses Theorem 6.1(i) only for pairs of a point A = {x} and a set
B ⊂ B(x, π (N − 1)/K), so that it is consistent with Remark 5.2(b). The following
proof is due to [Oh1].
Theorem 6.5 (Generalized Bonnet-Myers diameter bound) Suppose that a metric measure space (X, d, m) satisﬁes CD(K, N ) with K > 0 and N ∈ (1, ∞). Then we
have the following:√(i) It holds that diam X ≤ π (N − 1)/K.
√(ii) Each x ∈ X has at most one point of distance π (N − 1)/K from x.
Proof. It is enough to consider the case K = N − 1 thanks to the scaling property
Remark 5.2(a).
(i) Suppose that there is a pair of points x, y ∈ X with d(x, y) > π, set δ := d(x, y) −π > 0 and take a minimal geodesic γ : [0, π + δ] −→ X from x to y. Choosing a diﬀerent
point on γ if necessary, we can assume δ < π/2. For ε ∈ (0, δ), we apply Theorem 6.1(i)between {γ(δ + ε)} and B(y, ε) with t = (π − δ − ε)/π and obtainm(Zt ({γ(δ + ε)}, B(y, ε)))≥ tN
infr∈(π−2ε,π)m(B(y, ε))(sin(tr)t sin r

)N −1

(
)N −1
sin(t(π − 2ε))=t.
sin(π − 2ε)Then it follows from t(π − 2ε) ≤ tπ = π − δ − ε that()N −1m(Zt ({γ(δ + ε)}, B(y, ε)))π − δ − ε sin(δ + ε)≥→∞m(B(y, ε))π
sin 2ε(6.3)

as ε tends to zero. Given z ∈ B(y, ε), we take a minimal geodesic η : [0, 1] −→ X fromγ(δ + ε) to z (see Figure 9), and derive from the triangle inequality that
(
)
(
)
{ (
)
}
d γ(δ + ε), η(t) = td γ(δ + ε), z < t d γ(δ + ε), y + ε = π − δ − ε.
Moreover, we deduce from d(γ(δ + ε), z) < π that
(
)
(
)
(
)
d x, η(t) ≥ d(x, z) − d z, η(t) > π + δ − ε − (1 − t)d γ(δ + ε), z> π + δ − ε − (1 − t)π = π − 2ε.
Thus we have()()
Zt {γ(δ + ε)}, B(y, ε) ⊂ B γ(δ + ε), π − δ − ε \ B(x, π − 2ε)⊂ B(x, π) \ B(x, π − 2ε).
Combining this with (6.3), we concludem(B(x, π) \ B(x, π − 2ε))= ∞.
ε↓0m(B(y, ε))

lim

31

B(y, ε)γ(δ + ε)

γ

x
η

η(t)

z

S(x, π − 2ε)S(γ(δ + ε), π − δ − ε)Figure 9
Furthermore, (6.2) and Theorem 6.3 show that∫π())
sinN −1 r dr (m B(x, π) \ B(x, π − 2ε) ≤ ∫ π−2ε
m
B(x,
π
−
2ε)
π−2ε
sinN −1 r dr0
∫ 2ε N −1)()
sinr dr (0m B(x, π − 2ε) ≤ m B(x, 2ε) .
= ∫ π−2ε
N −1
sin
r dr
0
Hence we have, again due to Theorem 6.3 (with K = 0),
(
)
(
)
(
)
m B(x, π) \ B(x, π − 2ε) ≤ m B(x, 2ε) ≤ 2N m B(x, ε) .

(6.4)

Therefore we obtain limε↓0 m(B(x, ε))/m(B(y, ε)) = ∞. This is a contradiction because
we can exchange the roles of x and y.
(ii) We ﬁrst see that m(S(x, π)) = 0, where S(x, π) :=∪k {y ∈ X | d(x, y) = π}. Givenksmall ε > 0, take {xi }i=1 ⊂ S(x, 2ε) such that S(x, 2ε) ⊂ i=1 B(xi , 2ε) and d(xi , xj ) ≥ 2εholds if i 6= j. Then, for any y ∈ S(x, π), there is some xi so that d(y, xi ) < (π −2ε)+2ε =π, while d(y, xi ) ≥ π − 2ε holds in general. Thus we have()

m S(x, π) ≤ m(∪k)
B(xi , π) \ B(xi , π − 2ε)

i=1

≤

k
∑

(
)
m B(xi , π) \ B(xi , π − 2ε) .

i=1Then it follows from (6.4) that
k
k
∑
) ∑
(
)
(
)
N
m S(x, π) ≤m B(xi , 2ε) ≤ 2
m B(xi , ε)(i=1

= 2N m(∪k)
B(xi , ε)

i=1

32

i=1

(
)
≤ 2N m B(x, 3ε) .

Letting ε go to zero shows m(S(x, π)) = 0.
Now we suppose that there are two mutually distinct points y, z ∈ X such thatd(x, y) = d(x, z) = π. On the one hand, we derive from (6.4) that()()
m B(x, r) ≥ m(B(x, π) \ B x, π − r)
for r ∈ (0, π/2). On the other hand, as B(y, r) ⊂ X \ B(x, π − r) and m(S(x, π)) = 0, we
ﬁnd
(
)
(
)
m B(y, r) ≤ m B(x, π) \ B(x, π − r) .
Hence we obtain m(B(x, r)) ≥ m(B(y, r)) and similarly m(B(y, r)) ≥ m(B(x, r)). Thisimplies
(
)
(
)
(
)
(
)
m B(x, r) = m B(y, r) = m B(z, r) = m B(x, π) \ B(x, π − r) .
Then we have, for ε < d(y, z)/2,
(
)
(
)
(
)
(
)
2m B(x, ε) = m B(y, ε) + m B(z, ε) = m B(y, ε) ∪ B(z, ε)()()≤ m B(x, π) \ B(x, π − ε) = m B(x, ε) .
2This is obviously a contradiction.

For (X, d, m) satisfying CD(K, ∞) with K > 0, though X is not necessarily bounded(see Example 4.5), we can verify that m(X) is ﬁnite ([St3, Theorem 4.26]).

6.2

Maximal diameter

In Riemannian geometry, it is well known that the maximal diameter π among (unweighted) Riemannian manifolds of Ricci curvature ≥ n − 1 is achieved only by the unit
sphere Sn . In our case, however, orbifolds Sn /Γ can also have the maximal diameter.
Hence what we can expect is a decomposition into a spherical suspension (in some sense).
Due to the scaling property (Remark 5.2(a)), we consider only the case of K = N − 1 > 0.
See [Oh2] for more precise discussion of the following theorem, and Subsection 8.1 for thedeﬁnition of the non-branching property.
Theorem 6.6 Assume that (X, d, m) is non-branching and satisﬁes CD(N − 1, N ) forsome N ∈ (1, ∞) as well as diam X = π. Then (X, m) is the spherical suspension ofsome topological measure space.
Outline of proof. Fix xN , xS ∈ X with d(xN , xS ) = π. Then it follows from (6.4) that()()
m B(xN , r) + m B(xS , π − r) = m(X)for all r ∈ (0, π). This together with the non-branching property shows that, for anyz ∈ X \ {xN , xS }, there exists a unique minimal geodesic from xN to xS passing throughz.

33

Now we introduce the set Y consisting of unit speed minimal geodesics from xN to
xS , and equip it with the distance()
dY (γ1 , γ2 ) := sup dX γ1 (t), γ2 (t) .
0≤t≤πWe consider SY := (Y × [0, π])/ ∼, where (γ1 , t1 ) ∼ (γ2 , t2 ) holds if t1 = t2 = 0 or
t1 = t2 = π. We equip SY with the topology naturally induced from dY . Then the map
Ψ : SY 3 (γ, t) 7−→ γ(t) ∈ X is well-deﬁned and continuous. Deﬁne the mesures ν on Y
and ω on SY by
{∫ π
}−1
(
)
N −1
ν(W ) :=sint dtm Ψ(W × [0, π]) ,0dω := dν × (sinN −1 t dt).
Then one can prove that (SY, ω) is regarded as the spherical suspension of (Y, ν) as topological measure spaces, and that Ψ : (SY, ω) −→ (X, m) is homeomorphic and measurepreserving. We use the non-branching property for the continuity of Ψ−1 .
2

6.3

Open questions

We close the section with a list of open questions.
(A) (Beyond Theorem 6.6) There is room for improvement of Theorem 6.6: Is thenon-branching property necessary? Can one say anything about the relation between the
distances of SY and X? Does (Y, dY , ν) satisfy CD(N − 2, N − 1)? We do not know any
counterexample.
If X is an n-dimensional Alexandrov space of curvature ≥ 1 with diam X = π, thenit is isometric to the spherical suspension of some (n − 1)-dimensional Alexandrov spaceof curvature ≥ 1. It is generally diﬃcult to derive something about distance from thecurvature-dimension condition.
(B) (Extremal case of Lichnerowicz inequality) Related to Theorem 6.5, we know thefollowing ([LV1, Theorem 5.34]).
Theorem 6.7 (Generalized Lichnerowicz inequality) Assume that (X, d, m) satisﬁes CD(K, N ) for some K > 0 and N ∈ (1, ∞). Then we have
∫
∫
N −1
2
f dm ≤|∇− f |2 dm
(6.5)
KN X
X
∫
for any Lipschitz function f : X −→ R with X f dm = 0.
Here |∇− f | is the generalized gradient of f deﬁned by|∇− f |(x) := lim supy→x

max{f (x) − f (y), 0}.
d(x, y)The proof is done via careful calculation using (5.4) for SN between m(X)−1 · m and its
perturbation (1 + εf )m(X)−1 · m. The inequality (6.5) means that the lowest positive34

eigenvalue of the Laplacian is larger than or equal to KN/(N − 1). The constant (N −1)/KN in (6.5) is sharp. Moreover, in Riemannian geometry, it is known that the best
constant with N = dim M is achieved only by spheres.
In our general setting, it is not√ known whether the best constant is achieved only
by spaces of maximal diameter π (N − 1)/K. If so, then Theorem 6.6 provides us adecomposition into a spherical suspension (for non-branching spaces).
(C) (Splitting) In Riemannian geometry, Cheeger and Gromoll’s [CG] celebrated theorem asserts that, if a complete Riemannian manifold of nonnegative Ricci curvatureadmits an isometric embedding of the real line R ,→ M , then M isometrically splits oﬀ
R, namely M is isometric to a product space M 0 × R, where M 0 again has the nonnegative Ricci curvature. We can repeat this procedure if M 0 contains a line. This is anextremely deep theorem, and its generalization is a key tool of Cheeger and Colding’s
seminal work [CC] (see Further Reading of Section 5).
Kuwae and Shioya [KS3] consider (weighted) Alexandrov spaces of curvature ≥ −1with nonnegative Ricci curvature in terms of the measure contraction property (see Subsection 8.3). They show that, if such an Alexandrov space contains an isometric copyof the real line, then it splits oﬀ R as topological measure spaces (compare this with
Theorem 6.6). This is recently strengthened into an isometric splitting by [ZZ] under aslightly stronger notion of Ricci curvature bound in terms of Petrunin’s second variation
formula ([Pe1]).
For general metric measure spaces satisfying CD(0, N ), the isometric splitting is false
because n-dimensional Banach spaces satisfy CD(0, n) (Theorem 7.3) and do not split ingeneral. The homeomorphic, measure-preserving splitting could be true, but it is open
even for non-branching spaces.
(D) (Lévy-Gromov isoperimetric inequality) Another challenging problem is to show
(some appropriate variant of) the Lévy-Gromov isoperimetric inequality using optimaltransport. Most known proofs in the Riemannian case appeal to the deep existence andregularity theory of minimal surfaces which can not be expected in singular spaces.
For instance, let us consider the isoperimetric proﬁle IM : (0, m(M )) −→ (0, ∞) ofa weighted Riemannian manifold (M, g, m) with m = e−ψ volg , i.e., IM (V ) is the least
perimeter of sets with volume V . Bayle [Bay] shows that the diﬀerential inequality
N/(N −1) 00

(IM

) ≤−

KN 1/(N −1)−1
I
N −1 M(6.6)

holds if RicN ≥ K, which immediately implies the corresponding Lévy-Gromov isoperimetric inequality
IM (t · m(M ))≥ IK,N (t)m(M )for t ∈ [0, 1], where IK,N is the isoperimetric proﬁle of the N -dimensional space form of
constant sectional curvature K/(N − 1) equipped with the normalized measure (extended
to non-integer N numerically). The concavity estimate (6.6) seems to be related to the
Brunn-Minkowski inequality, however, Bayle’s proof of (6.6) is based on the variational
formulas of minimal surfaces (see also [Mo, Chapter 18]). More analytic approach could
work in metric measure spaces, but we need a new idea for it.
35

Further Reading The generalized Brunn-Minkowski inequality (Theorem 6.1) is essentially contained in the proof of [vRS, Theorem 1.1] for N = ∞, and due to Sturm [St4]for N < ∞ (also the Brascamp-Lieb inequality in [CMS1] implies it in the unweighted
Riemannian situation with N = n). It is used as a key tool in the proof of the derivationof the Ricci curvature bound from the curvature-dimension condition (see Theorem 4.6).
Some more related interpolation inequalities can be found in [CMS1], these all were neweven for Riemannian manifolds.
The relation between CD(K, ∞) and functional inequalities such as the Talagrand,
logarithmic Sobolev and the global Poincaré inequalities are studied by Otto and Villani[OV] and Lott and Villani [LV2, Section 6]. We refer to [Ol], [BoS] for related work on
discrete spaces (see also Subsection 7.3(B)), and to [St2], [Vi2, Chapter 25], [OT] for the
relation between variants of these functional inequalities and the displacement convexityof generalized entropies. Theorems 6.5 and 6.6 are due to [Oh1] and [Oh2], where theproof is given in terms of the measure contraction property (see Subsection 8.3).

7

The curvature-dimension condition in Finsler geometry

In this section, we demonstrate that almost everything so far works well also in theFinsler setting. In fact, the equivalence between RicN ≥ K and CD(K, N ) is extended byintroducing an appropriate notion of the weighted Ricci curvature. Then we explain why
this is signiﬁcant and discuss two potential applications. We refer to [BCS] and [Sh2] forthe fundamentals of Finsler geometry, and the main reference of the section is [Oh5].

7.1

A brief introduction to Finsler geometry

Let M be an n-dimensional connected C ∞ -manifold. Given a local coordinate (xi )ni=1 onan open set U ⊂ M , we always consider the coordinate (xi , v i )ni=1 on T U given by
∂ ¯¯
v=
v
¯ ∈ Tx M.
∂xi x
i=1
n
∑

i

Deﬁnition 7.1 (Finsler structures) A C ∞ -Finsler structure is a nonnegative function
F : T M −→ [0, ∞) satisfying the following three conditions:(1) (Regularity) F is C ∞ on T M \ 0, where 0 stands for the zero section;(2) (Positive homogeneity) F (λv) = λF (v) holds for all v ∈ T M and λ ≥ 0;(3) (Strong convexity) Given a local coordinate (xi )ni=1 on U ⊂ M , the n × n matrix
)n
( 2 2
(
)n
1 ∂ (F )(7.1)(v)
gij (v) i,j=1 :=2 ∂v i ∂v ji,j=1
is positive-deﬁnite for all v ∈ Tx M \ 0, x ∈ U .
36

In other words, each F |Tx M is a C ∞ -Minkowski norm (see Example 7.4(a) below for the
precise deﬁnition) and it varies C ∞ -smoothly also in the horizontal direction. We remarkthat the homogeneity (2) is imposed only in the positive direction, so that F (−v) 6=
F (v) is allowed. The positive-deﬁnite symmetric matrix (gij (v))ni,j=1 in (7.1) deﬁnes the
Riemannian structure gv on Tx M through
(∑
n
n
n
¯ ∑
¯ )
∑
j ∂ ¯
i ∂ ¯
gij (v)v1i v2j .
(7.2)
gv
v1 i ¯ ,v2 j ¯ :=∂x x j=1 ∂x x
i=1
i,j=1
Note that F (v)2 = gv (v, v). If F is coming from a Riemannian structure, then gv always
coincides with the original Riemannian metric. In general, the inner product gv is regarded
as the best approximation of F in the direction v. More precisely, the unit spheres of F
and gv are tangent to each other at v/F (v) up to the second order (that is possible thanks
to the strong convexity, see Figure 10).
6F (·) = 1
gv (·, ·) = 1

v/F (v) --Figure 10
The distance between x, y ∈ M is naturally deﬁned by
}
{∫ 1
¯
¯
1
d(x, y) := inf
F (γ̇) dt ¯ γ : [0, 1] −→ M, C , γ(0) = x, γ(1) = y .
0

One remark is that the nonsymmetry d(x, y) 6= d(y, x) may come up as F is only positively
homogeneous. Thus it is not totally correct to call d a distance, it might be called costor action as F is a sort of Lagrangian cost function. Another remark is that the functiond(x, ·)2 is C 2 at the origin x if and only if F |Tx M is Riemannian. Indeed, the squared
norm | · |2 of a Banach (or Minkowski) space (Rn , | · |) is C 2 at 0 if and only if it is an
inner product.
A C ∞ -curve γ : [0, l] −→ M is called a geodesic if it has constant speed (F (γ̇) ≡ c ∈[0, ∞)) and is locally minimizing (with respect to d). The reverse curve γ̄(t) := γ(l − t)
is not necessarily a geodesic. We say that (M, F ) is forward complete if any geodesicγ : [0, ε] −→ M is extended to a geodesic γ : [0, ∞) −→ M . Then any two points
x, y ∈ M are connected by a minimal geodesic from x to y.
37

7.2

Weighted Ricci curvature and the curvature-dimension condition

We introduced distance and geodesics in a natural (metric geometric) way, but the definition of curvature is more subtle. The ﬂag and Ricci curvatures on Finsler manifolds,
corresponding to the sectional and Ricci curvatures in Riemannian geometry, are deﬁned
via some connection as in the Riemannian case. The choice of connection is not uniquein the Finsler setting, nevertheless, all connections are known to give rise to the same
curvature. In these notes, however, we shall follow Shen’s idea [Sh2, Chapter 6] of introducing the ﬂag curvature using vector ﬁelds and corresponding Riemannian structures(via (7.2)). This intuitive description is not only geometrically understandable, but also
useful and inspiring.
Fix a unit vector v ∈ Tx M ∩ F −1 (1), and extend it to a C ∞ -vector ﬁeld V on an
open neighborhood U of x in such a way that every integral curve of V is geodesic. In
particular, V (γ(t)) = γ̇(t) along the geodesic γ : (−ε, ε) −→ M with γ̇(0) = v. Using(7.2), we equip U with the Riemannian structure gV . Then the ﬂag curvature K(v, w) ofv and a linearly independent vector w ∈ Tx M coincides with the sectional curvature withrespect to gV of the 2-plane v ∧ w spanned by v and w. Similarly, the Ricci curvature
Ric(v) of v (with respect to F ) coincides with the Ricci curvature of v with respect to gV .
This contains the fact that K(v, w) is independent of the choice of the extension V of v.
We remark that K(v, w) depends not only on the ﬂag v ∧ w, but also on the choice of the
ﬂagpole v in the ﬂag v ∧ w. In particular, K(v, w) 6= K(w, v) may happen.
As for measure, on Finsler manifolds, there is no constructive measure as good as the
Riemannian volume measure. Therefore, as the theory of weighted Riemannian manifolds,
we equip (M, F ) with an arbitrary positive C ∞ -measure m on M . Now, the weighted Ricci
curvature is deﬁned as follows ([Oh5]). We extend given a unit vector v ∈ Tx M to a C ∞ vector ﬁeld V on a neighborhood U 3 x such that every integral curve is geodesic (or itis suﬃcient to consider only the tangent vector ﬁeld γ̇ of the geodesic γ : (−ε, ε) −→ Mwith γ̇(0) = v), and decompose m as m = e−Ψ(V ) volgV on U . We remark that the weightΨ is not a function on M , but a function on the unit tangent sphere bundle SM ⊂ T M .
For simplicity, we set∂v Ψ :=d(Ψ ◦ γ̇)(0),dt

∂v2 Ψ :=d2 (Ψ ◦ γ̇)
(0).
dt2

(7.3)

Deﬁnition 7.2 (Weighted Ricci curvature of Finsler manifolds) For N ∈ [n, ∞]and a unit vector v ∈ Tx M , we deﬁne{Ric(v) + ∂v2 Ψ if ∂v Ψ = 0,
(1) Ricn (v) :=−∞otherwise;(2) RicN (v) := Ric(v) + ∂v2 Ψ −(∂v Ψ)2
for N ∈ (n, ∞);N −n(3) Ric∞ (v) := Ric(v) + ∂v2 Ψ.
In other words, RicN (v) of F is RicN (v) of gV (recall Deﬁnition 4.4), so that thiscurvature coincides with RicN in weighted Riemannian manifolds. We remark that the38quantity ∂v Ψ coincides with Shen’s S-curvature (also called the mean covariance or meantangent curvature, see [Sh1], [Sh2], [Sh3]). Therefore bounding Ricn from below makes
sense only when the S-curvature vanishes everywhere. This curvature enables us to extend
Theorem 4.6 to the Finsler setting ([Oh5]). Therefore all results in the theory of curvaturedimension condition are applicable to general Finsler manifolds.
Theorem 7.3 A forward complete Finsler manifold (M, F, m) equipped with a positive
C ∞ -measure m satisﬁes CD(K, N ) for some K ∈ R and N ∈ [n, ∞] if and only if
RicN (v) ≥ K holds for all unit vectors v ∈ T M .
We remark that, in the above theorem, the curvature-dimension condition is appropriately extended to nonsymmetric distances. The proof of Theorem 7.3 follows the sameline as the Riemannian case, however, we should be careful about nonsymmetric distanceand need some more extra discussion due to the fact that the squared distance functiond(x, ·)2 is only C 1 at x.
We present several examples of Finsler manifolds. The ﬂag and Ricci curvatures arecalculated in a number of situations, while the weighted Ricci curvature is still relatively
much less investigated.
Example 7.4 (a) (Banach/Minkowski spaces with Lebesgue measures) A Minkowski
norm | · | on Rn is a nonsymmetric generalization of usual norms. That is to say, | · | isa nonnegative function on Rn satisfying the positive homogeneity |λv| = λ|v| for v ∈ Rn
and λ > 0; the convexity |v + w| ≤ |v| + |w| for v, w ∈ Rn ; and the positivity |v| > 0 for
v 6= 0. Note that the unit ball of | · | is a convex (but not necessarily symmetric to theorigin) domain containing the origin in its interior (see Figure 10, where F is a Minkowskinorm).
A Banach or Minkowski norm | · | which is uniformly convex and C ∞ on Rn \ {0}induces a Finsler structure in a natural way through the identiﬁcation between Tx Rn and
Rn . Then (Rn , | · |, voln ) has the ﬂat ﬂag curvature. Hence a Banach or Minkowski space(Rn , | · |, voln ) satisﬁes CD(0, n) by Theorem 7.3 for C ∞ -norms, and by Theorem 5.6 via
approximations for general norms.
(b) (Banach/Minkowski spaces with log-concave measures) A Banach or Minkowski
space (Rn , | · |, m) equipped with a measure m = e−ψ voln such that ψ is K-convex withrespect to | · | satisﬁes CD(K, ∞). Here the K-convexity means that()
K
ψ (1 − t)x + ty ≤ (1 − t)ψ(x) + tψ(y) − (1 − t)t|y − x|22
holds for all x, y ∈ Rn and t ∈ [0, 1]. This is equivalent to ∂v2 ψ ≥ K (in the sense of (7.3))
if | · | and ψ are C ∞ (on Rn \ {0} and Rn , respectively). Hence CD(K, ∞) again follows
from Theorem 7.3 together with Theorem 5.6.
2
In particular, a Gaussian type space (Rn , | · |, e−|·| /2 voln ) satisﬁes CD(0, ∞) independently of n. It also satisﬁes CD(K, ∞) for some K > 0 if (and only if) it is 2-uniformly
convex in the sense that | · |2 /2 is C −2 -convex for some C ≥ 1 (see [BCL] and [Oh4]),−2
and then√ K = C . For instance, `p -spaces with p ∈ (1, 2] are 2-uniformly convex with
C = 1/ p − 1, and hence satisﬁes CD(p − 1, ∞). Compare this with Example 4.5.
39

(c) (Randers spaces) A Randers space (M, F ) is a special kind of Finsler manifold suchthat√
F (v) = g(v, v) + β(v)for some Riemannian metric g and a one-form β. We suppose that |β(v)|2 < g(v, v)
unless v = 0, then F is indeed a Finsler structure. Randers spaces are important inapplications and reasonable for concrete calculations. In fact, we can see by calculationthat S(v) = ∂v Ψ ≡ 0 holds if and only if β is a Killing form of constant length as well
as m is the Busemann-Hausdorﬀ measure (see [Oh7], [Sh2, Section 7.3] for more details).
This means that there are many Finsler manifolds which do not admit any measures ofRicn ≥ K > −∞, and then we must consider RicN for N > n.
(d) (Hilbert geometry) Let D ⊂ Rn be a bounded open set with smooth boundarysuch that its closure D is strictly convex. Then the associated Hilbert distance is deﬁnedby()
kx1 − x02 k · kx2 − x01 kd(x1 , x2 ) := log
kx1 − x01 k · kx2 − x02 k
for distinct x1 , x2 ∈ D, where k · k is the standard Euclidean norm and x01 , x02 are intersections of ∂D and the line passing through x1 , x2 such that x0i is on the side of xi .
Hilbert geometry is known to be realized by a Finsler structure with constant negative
ﬂag curvature. However, it is still unclear if it carries a (natural) measure for which thecurvature-dimension condition holds.
(e) (Teichmüller space) Teichmüller metric on Teichmüller space is one of the most
famous Finsler structures in diﬀerential geometry. It is known to be complete, while theWeil-Petersson metric is incomplete and Riemannian. The author does not know any
investigation concerned with the curvature-dimension condition of Teichmüller space.

7.3

Remarks and potential applications

Due to celebrated work of Cheeger and Colding [CC], we know that a (non-Hilbert) Banach
space can not be the limit space of a sequence of Riemannian manifolds (with respect to themeasured Gromov-Hausdorﬀ convergence) with a uniform lower Ricci curvature bound.
Therefore the fact that Finsler manifolds satisfy the curvature-dimension condition meansthat it is too weak to characterize limit spaces of Riemannian manifolds. This should becompared with the following facts.
(I) A Banach space can be an Alexandrov space only if it happens to be a Hilbert
space (and then it has the nonnegative curvature);(II) It is not known if all Alexandrov spaces X of curvature ≥ k can be approximatedby a sequence of Riemannian manifolds {Mi }i∈N of curvature ≥ k 0 .
We know that there are counterexamples to (II) if we impose the non-collapsing
condition dim Mi ≡ dim X (see [Ka]), but the general situation admitting collapsing(dim Mi > dim X) is still open and is one of the most important and challenging questions in Alexandrov geometry. Thus the curvature-dimension condition is not as goodas the Alexandrov-Toponogov triangle comparison condition from the purely Riemannian
geometric viewpoint.

40

From a diﬀerent viewpoint, Cheeger and Colding’s observation means that the familyof Finsler spaces is properly much wider than the family of Riemannian spaces. Therefore
the validity of the curvature-dimension condition for Finsler manifolds opens the door tobroader applications. Here we mention two of them.
(A) (The geometry of Banach spaces) Although their interested spaces are common tosome extent, there is almost no connection between the geometry of Banach spaces andFinsler geometry (as far as the author knows). We believe that our diﬀerential geometrictechnique would be useful in the geometry of Banach spaces. For instance, Theorem 7.3(together with Theorem 5.6) could recover and generalize Gromov and Milman’s normal
concentration of unit spheres in 2-uniformly convex Banach spaces (see [GM2] and [Le,Section 2.2]). To be precise, as an application of Theorem 7.3, we know the normal
concentration of Finsler manifolds such that Ric∞ goes to inﬁnity (see [Oh5], and [GM1],[Le, Section 2.2] for the Riemannian case). This seems to imply the concentration of unit
spheres mentioned above.
(B) (Approximations of graphs) Generally speaking, Finsler spaces give much better
approximations of graphs than Riemannian spaces, when we impose a lower Ricci curvature bound. For instance, Riemannian spaces into which the Zn -lattice is nearly isometrically embedded should have very negative curvature, while the Zn -lattice is isometricallyembedded in ﬂat `n1 . This kind of technique seems useful for investigating graphs with
Ricci curvature bounded below (in some sense), and provides a diﬀerent point of view onvariants of the curvature-dimension condition for discrete spaces (see, e.g., [Ol], [BoS]).
Further Reading We refer to [BCS] and [Sh2] for the fundamentals of Finsler geometry
and important examples. The interpretation of the ﬂag curvature using vector ﬁelds
can be found in [Sh2, Chapter 6]. We also refer to [Sh1] and [Sh3] for the S-curvatureand its applications including a volume comparison theorem diﬀerent from Theorem 6.3(which has some topological applications). The S-curvature of Randers spaces and thecharacterization of its vanishing (Example 7.4(c)) are studied in [Sh2, Section 7.3] and[Oh7].
Deﬁnition 7.2 and Theorem 7.3 are due to [Oh5], while the weight function Ψ on SM
has already been considered in the deﬁnition of S-curvature. See also [OhS] for related
work concerning heat ﬂow on Finsler manifolds, and [Oh6] for a survey on these subjects.
The curvature-dimension condition CD(0, n) of Banach spaces (Example 7.4(a)) is ﬁrst
demonstrated by Cordero-Erausquin (see [Vi2, page 908]).

8

Related topics

We brieﬂy comment on further related topics.

8.1

Non-branching spaces

We say that a geodesic space (X, d) is non-branching if geodesics do not branch in thesense that each quadruple of points z, x0 , x1 , x2 ∈ X with d(x0 , x1 ) = d(x0 , x2 ) = 2d(z, xi )(i = 0, 1, 2) must satisfy x1 = x2 . This is a quite useful property. For instance, in sucha space satisfying CD(K, N ) for some K and N , a.e. x ∈ X has unique minimal geodesic
41

from x to a.e. y ∈ X ([St4, Lemma 4.1]). Therefore we can localize the inequality (5.3),
and then (5.3) for single U = SN implies that for all U ∈ DC N (see Remark 5.2(e), (f),[St4, Proposition 4.2]). There are some more results known only in non-branching spaces(see, e.g., [St4, Section 4], [FV] and also Subsection 8.3 below).
Riemannian (or Finsler) manifolds and Alexandrov spaces are clearly non-branching.
However, as n-dimensional Banach and Minkowski spaces satisfy CD(0, n), the curvaturedimension condition does not prevent the branching phenomenon. One big open problem
after Cheeger and Colding’s work [CC] is whether any limit space of Riemannian manifolds
with a uniform lower Ricci curvature bound is non-branching or not.

8.2

Alexandrov spaces

As was mentioned in Remark 3.1, Alexandrov spaces are metric spaces whose sectional
curvature is bounded from below in terms of the triangle comparison property (see [BGP],[OtS], [BBI, Chapters 4, 10] for more details). One interesting fact is that a compact
geodesic space (X, d) is an Alexandrov space of nonnegative curvature if and only if theWasserstein space (P(X), dW2 ) over it is an Alexandrov space of nonnegative curvature([St3, Proposition 2.10], [LV2, Theorem A.8]). This is a metric geometric explanation of
Otto’s formal calculation of the sectional curvature of (P2 (Rn ), dW
2 ) ([Ot]). We remarkthat this relation can not be extended to positive or negative curvature bounds. In fact,
if (X, d) is not an Alexandrov space of nonnegative curvature, then (P(X), dW2 ) is not an
Alexandrov space of curvature ≥ k even for negative k ([St3, Proposition 2.10]). Optimaltransport in Alexandrov spaces is further studied in [Be], [Oh3], [Sav], [Gi] and [GO].
Since the Ricci curvature is the trace of the sectional curvature, it is natural to expectthat Alexandrov spaces satisfy the curvature-dimension condition. Petrunin [Pe3] recently
claims that it is indeed the case for K = 0, and is extended to the general case K 6= 0
by [ZZ]. They use the second variation formula in [Pe1] and the gradient ﬂow techniquedeveloped in [PP] and [Pe2], instead of calculations as in Sections 3, 4 involving Jacobi
ﬁelds.

8.3

The measure contraction property

For K ∈ R and N ∈ (1, ∞), a metric measure space is said to satisfy the measure contraction property MCP(K, N ) if the Bishop inequality (3.3) holds in an appropriate sense.
More precisely, MCP(K, N ) for (X, d, m) means that any x ∈ X admits a measurable
map Φ : X −→ Γ(X) satisfying e0 ◦ Φ ≡ x, e1 ◦ Φ = IdX and
(
)
(
)
t
d(x, y) dm(y)dm ≥ (et ◦ Φ)] tN βK,Nfor all t ∈ (0, 1) as measures (compare this with Theorem 6.1(i)). As we mentioned in
Further Reading in Section 3, this kind of property was suggested in [CC, I, Appendix 2]and [Gr, Section 5.I], and systematically studied in [Oh1], [Oh2] and [St4, Sections 5, 6].
Some variants have been also studied in [KS1] and [St1] before them.
MCP(K, N ) can be regarded as the curvature-dimension condition CD(K, N ) applied
only for each pair of a Dirac measure and a uniform distribution on a set, and CD actually42

implies MCP in non-branching spaces. It is known that Alexandrov spaces satisfy MCP(see [Oh1], [KS2]). For n-dimensional (unweighted) Riemannian manifolds, MCP(K, n) is
equivalent to Ric ≥ K, however, MCP(K, N ) with N > n does not imply Ric ≥ K. In fact,
a suﬃciently small ball in Rn equipped with the Lebesgue measure satisﬁes MCP(1, n+1).
This is one drawback of MCP. On the other hand, an advantage of MCP is its simpleness,there are several facts known for MCP and unknown for CD. We shall compare theseproperties in more details.
(A) (Product spaces (L2 -tensorization property)) If (Xi , di , mi ) satisﬁes MCP(Ki , Ni )for i = 1, 2, then the product metric measure space (X1 × X2 , d1 × d2 , m1 × m2 ) satisﬁes
MCP(min{K1 , K2 }, N1 + N2 ) ([Oh2]). The analogous property for CD is known only formin{K1 , K2 } = 0 or N1 + N2 = ∞ in non-branching spaces ([St3]).
Recently, Bacher and Sturm [BaS1] introduce a slightly weaker variant of CD, called the
reduced curvature-dimension condition CD∗ (recall (4.15) in Further Reading of Section 4).
They show that CD∗ enjoys the tensorization property if the spaces in consideration are
non-branching.
(B) (Euclidean cones) If (X, d, m) satisﬁes MCP(N − 1, N ), then its Euclidean cone
(CX, dCX , mCX ) deﬁned by()
CX := X × [0, ∞) / ∼, (x, 0) ∼ (y, 0),√()
dCX (x, s), (y, t) := s2 + t2 − 2st cos d(x, y),
dmCX := dm × (tN dt)
satisﬁes MCP(0, N + 1) ([Oh2]). This is recently established for the curvature-dimension
condition by [BaS2] in the case where (X, d, m) is Riemannian.
(C) (Local-to-global property) Sturm [St3] shows that, if (X, d, m) is non-branching
and if every point in X admits an open neighborhood on which CD(K, ∞) holds, then thewhole space (X, d, m) globally satisﬁes CD(K, ∞). In other words, CD(K, ∞) is a local
condition as is the Ricci curvature bound on a Riemannian manifold. The same holds
true also for CD(0, N ) with N < ∞. It is shown in [BaS1] that CD∗ (K, N ) satisﬁes the
local-to-global property for general K ∈ R and N ∈ (1, ∞), however, it is still open andunclear if CD(K, N ) for K 6= 0 and N < ∞ is a local condition.
In contrast, the local-to-global property is known to be false for MCP. As we mentioned
above, suﬃciently small balls in Rn satisfy MCP(1, n + 1), while the entire space Rn does
not satisfy it by virtue of the Bonnet-Myers diameter bound (Theorem 6.5).

References[AGS]

L. Ambrosio, N. Gigli and G. Savaré, Gradient ﬂows in metric spaces and in thespace of probability measures, Birkhäuser Verlag, Basel, 2005.

[BaS1] K. Bacher and K.-T. Sturm, Localization and tensorization properties of thecurvature-dimension condition for metric measure spaces, J. Funct. Anal. 259(2010), 28–56.
[BaS2] K. Bacher and K.-T. Sturm, Ricci bounds for Euclidean and spherical cones,
Preprint (2010). Available at arXiv:1103.0197
43

[BE]

D. Bakry and M. Émery, Diﬀusions hypercontractives (French), Séminaire deprobabilités, XIX, 1983/84, 177–206, Lecture Notes in Math. 1123, Springer,
Berlin, 1985.

[BCL]

K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness
inequalities for trace norms, Invent. Math. 115 (1994), 463–482.

[Bal]

W. Ballmann, Lectures on spaces of nonpositive curvature. With an appendix by
Misha Brin, Birkhäuser Verlag, Basel, 1995.

[BCS]

D. Bao, S.-S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry,
Springer-Verlag, New York, 2000.

[Bay]

V. Bayle, Propriétés de concavité du proﬁl isopérimétrique et applications(French), Thèse de Doctorat, Institut Fourier, Université Joseph-Fourier, Grenoble, 2003.

[Be]

J. Bertrand, Existence and uniqueness of optimal maps on Alexandrov spaces,
Adv. Math. 219 (2008), 838–851.

[BoS]

A.-I. Bonciocat and K.-T. Sturm, Mass transportation and rough curvaturebounds for discrete spaces, J. Funct. Anal. 256 (2009), 2944–2966.

[Br]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued
functions, Comm. Pure Appl. Math. 44 (1991), 375–417.

[BBI]

D. Burago, Yu. Burago and S. Ivanov, A course in metric geometry, American
Mathematical Society, Providence, RI, 2001.

[BGP] Yu. Burago, M. Gromov and G. Perel’man, A. D. Alexandrov spaces with curvatures bounded below, Russian Math. Surveys 47 (1992), 1–58.
[Ch]

I. Chavel, Riemannian geometry. A modern introduction. Second edition, Cambridge University Press, Cambridge, 2006.

[CC]

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvaturebounded below. I, II, III, J. Diﬀerential Geom. 46 (1997), 406–480; ibid. 54(2000), 13–35; ibid. 54 (2000), 37–74.

[CE]

J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, Revised reprint of the 1975 original, AMS Chelsea Publishing, Providence, RI, 2008.

[CG]

J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative
Ricci curvature, J. Diﬀerential Geometry 6 (1971/72), 119–128.

[CMS1] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146(2001), 219–257.

44

[CMS2] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, PrékopaLeindler type inequalities on Riemannian manifolds, Jacobi ﬁelds, and optimaltransport, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), 613–635.
[FF]

A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds, Israel
J. Math. 175 (2010), 1–59.

[FG]

A. Figalli and N. Gigli, Local semiconvexity of Kantorovich potentials on noncompact manifolds, ESAIM Control Optim. Calc. Var. 17 (2011), 648–653.

[FV]

A. Figalli and C. Villani, Strong displacement convexity on Riemannian manifolds, Math. Z. 257 (2007), 251–259.

[Fu]

K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), 517–547.

[Ga]

R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.)
39 (2002), 355–405.

[Gi]

N. Gigli, On the inverse implication of Brenier-McCann theorems and the structure of (P2 (M ), W2 ), Methods Appl. Anal. 18 (2011), 127–158.

[GO]

N. Gigli and S. Ohta, First variation formula in Wasserstein spaces over compact
Alexandrov spaces, Canad. Math. Bull. 55 (2012), 723–735.

[Gr]

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces,
Birkhäuser, Boston, MA, 1999.

[GM1] M. Gromov and V. D. Milman, A topological application of the isoperimetricinequality, Amer. J. Math. 105 (1983), 843–854.
[GM2] M. Gromov and V. D. Milman, Generalization of the spherical isoperimetricinequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–
282.
[Ka]

V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom.
Funct. Anal. 12 (2002), 121–137.

[KS1]

K. Kuwae and T. Shioya, On generalized measure contraction property and energy functionals over Lipschitz maps, ICPA98 (Hammamet). Potential Anal. 15(2001), 105–121.

[KS2]

K. Kuwae and T. Shioya, Inﬁnitesimal Bishop-Gromov condition for Alexandrov
spaces, Adv. Stud. Pure Math. 57 (2010), 293–302.

[KS3]

K. Kuwae and T. Shioya, A topological splitting theorem for weighted Alexandrov
spaces, Tohoku Math. J. (2) 63 (2011), 59–76.

[Le]

M. Ledoux, The concentration of measure phenomenon, American MathematicalSociety, Providence, RI, 2001.
45

[Lo1]

J. Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment.
Math. Helv. 78 (2003), 865–883.

[Lo2]

J. Lott, Optimal transport and Ricci curvature for metric-measure spaces, Surveys in diﬀerential geometry XI, 229–257, Int. Press, Somerville, MA, 2007.

[LV1]

J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J.
Funct. Anal. 245 (2007), 311–333.

[LV2]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimaltransport, Ann. of Math. 169 (2009), 903–991.

[Mc1]

R. J. McCann, A convexity principle for interacting gases, Adv. Math. 128(1997), 153–179.

[Mc2]

R. J. McCann, Polar factorization of maps on Riemannian manifolds, Geom.
Funct. Anal. 11 (2001), 589–608.

[Mo]

F. Morgan, Geometric measure theory. A beginner’s guide. Fourth edition, Elsevier/Academic Press, Amsterdam, 2009.

[Oh1]

S. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805–828.

[Oh2]

S. Ohta, Products, cones, and suspensions of spaces with the measure contraction
property, J. Lond. Math. Soc. (2) 76 (2007), 225–236.

[Oh3]

S. Ohta, Gradient ﬂows on Wasserstein spaces over compact Alexandrov spaces,
Amer. J. Math. 131 (2009), 475–516.

[Oh4]

S. Ohta, Uniform convexity and smoothness, and their applications in Finsler
geometry, Math. Ann. 343 (2009), 669–699.

[Oh5]

S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Diﬀerential Equations 36 (2009), 211–249.

[Oh6]

S. Ohta, Optimal transport and Ricci curvature in Finsler geometry, Adv. Stud.
Pure Math. 57 (2010), 323–342.

[Oh7]

S. Ohta, Vanishing S-curvature of Randers spaces, Diﬀerential Geom. Appl. 29(2011), 174–178.

[OhS]

S. Ohta and K.-T. Sturm, Heat ﬂow on Finsler manifolds, Comm. Pure Appl.
Math. 62 (2009), 1386–1433.

[OT]

S. Ohta and A. Takatsu, Displacement convexity of generalized relative entropies,
Adv. Math. 228 (2011), 1742–1787.

[Ol]

Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal.
256 (2009), 810–864.
46

[OtS]

Y. Otsu, T. Shioya, The Riemannian structure of Alexandrov spaces, J. Diﬀerential Geom. 39 (1994), 629–658.

[Ot]

F. Otto, The geometry of dissipative evolution equations: the porous medium
equation, Comm. Partial Diﬀerential Equations 26 (2001), 101–174.

[OV]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links
with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), 361–400.

[PP]

G. Perel’man and A. Petrunin, Quasigeodesics and gradient curves
in Alexandrov spaces, Unpublished preprint (1994). Available at
http://www.math.psu.edu/petrunin/[Pe1]

A. Petrunin, Parallel transportation for Alexandrov space with curvaturebounded below, Geom. Funct. Anal. 8 (1998), 123–148.

[Pe2]

A. Petrunin, Semiconcave functions in Alexandrov’s geometry, Surveys in diﬀerential geometry. Vol. XI, 137–201, Surv. Diﬀer. Geom. 11, Int. Press, Somerville,
MA, 2007.

[Pe3]

A. Petrunin, Alexandrov meets Lott-Villani-Sturm, Münster J. Math. 4 (2011),
53–64.

[Qi]

Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), 235–242.

[vRS]

M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates,
entropy and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), 923–940.

[Sak]

T. Sakai, Riemannian geometry, Translated from the 1992 Japanese original by
the author. Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996.

[Sav]

G. Savaré, Gradient ﬂows and diﬀusion semigroups in metric spaces under lower
curvature bounds, C. R. Math. Acad. Sci. Paris 345 (2007), 151–154.

[Sh1]

Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry,
Adv. Math. 128 (1997), 306–328.

[Sh2]

Z. Shen, Lectures on Finsler geometry, World Scientiﬁc Publishing Co., Singapore, 2001.

[Sh3]

Z. Shen, Landsberg curvature, S-curvature and Riemann curvature, A sampler ofRiemann-Finsler geometry, 303–355, Math. Sci. Res. Inst. Publ., 50, Cambridge
Univ. Press, Cambridge, 2004.

[St1]

K.-T. Sturm, Diﬀusion processes and heat kernels on metric spaces, Ann. Probab.
26 (1998), 1–55.

47

[St2]

K.-T. Sturm, Convex functionals of probability measures and nonlinear diﬀusions
on manifolds, J. Math. Pures Appl. 84 (2005), 149–168.

[St3]

K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196(2006), 65–131.

[St4]

K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196(2006), 133–177.

[Vi1]

C. Villani, Topics in optimal transportation, American Mathematical Society,
Providence, RI, 2003.

[Vi2]

C. Villani, Optimal transport, old and new, Springer-Verlag, Berlin, 2009.

[We]

G. Wei, Manifolds with a lower Ricci curvature bound, Surveys in diﬀerential
geometry XI, 203–227, Int. Press, Somerville, MA, 2007.

[ZZ]

H.-C. Zhang and X.-P. Zhu, Ricci curvature on Alexandrov spaces and rigidity
theorems, Comm. Anal. Geom. 18 (2010), 503–553.

48