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Advanced Studies in Pure Mathematics ,pp. 1–20

Optimal transport and Ricci curvaturein Finsler geometryShin-ichi Ohta
Abstract.
This is a survey article on recent progress (in [Oh3], [OS]) of thetheory of weighted Ricci curvature in Finsler geometry. Optimal transport theory plays an impressive role as is developed in the Riemanniancase by Lott, Sturm and Villani.

§1.

Introduction

A Finsler manifold is a generalization of a Riemannian manifold
admitting its tangent spaces being Banach spaces or, more generally,
Minkowski spaces. Then a natural question is: Is there a canonicalmeasure associated with each Finsler manifold? Generally speaking,the answer is no. There are several constructive measures, such as the
Busemann-Hausdorﬀ measure and the Holmes-Thompson measure, eachof which is canonical in some sense (see [AT]). One strategy for dealing
with this diﬃculty is not to choose one speciﬁc measure and to consider an arbitrary measure in the ﬁrst place, like the theory of weighted
Riemannian manifolds.
In weighted Riemannian geometry, we decompose given a measure m
on a Riemannian manifold M into m = e−ψ volM using the Riemannianvolume element volM . Then the weight function ψ plays an importantrole, for instance, the Bakry-Émery tensor Ric + Hess ψ is known tobehave like the (inﬁnite dimensional) Ricci curvature of this weighted
space. In order to follow this line in our Finsler setting, we need areference measure like volM . Our idea is that the reference measuredoes not live in the manifold, but in the unit sphere bundle. Moreprecisely, we do not ﬁx one reference measure, but choose a reference
2000 Mathematics Subject Classiﬁcation. 53C60, 49Q20, 58J35.
Key words and phrases. Finsler geometry, optimal transport, Ricci curvature, comparison theorem.
Supported in part by the Grant-in-Aid for Young Scientists (B) 20740036.

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2

S. Ohtameasure for each given unit tangent vector. At this point we are indebted
to Shen’s extremely useful interpretation of ﬂag and Ricci curvatures.
Then it turns out that many results in (weighted) Riemannian geometry
involving Ricci curvature can be translated into Finsler geometry.
One of the highlights stems from the connection with optimal transport theory. The curvature-dimension condition and the N -Ricci curvature bound are equivalent to the lower weighted Ricci curvature bound(Theorem 4.2) as is established in the Riemannian case by Lott, Sturmand Villani. This theorem has fruitful applications in interpolation inequalities, functional inequalities and the concentration of measure phenomenon.
Moreover, beyond just a generalization of the Riemannian case, thisresult has potential applications in at least two directions. One directionis the approximation of new kinds of spaces. The curvature-dimension
condition is stable under the measured Gromov-Hausdorﬀ convergence,but even Banach spaces can not be approximated by Riemannian manifolds with a uniform lower Ricci curvature bound. Therefore the limitof Finsler manifolds covers much wider class of spaces than that of Riemannian manifolds. For instance, Cordero-Erausquin’s result on thecurvature-dimension condition of Banach spaces (see [Vi2, Theorem inpage 908]) is an immediate consequence of Theorem 4.2. The other direction is the connection with Banach space theory. Finsler geometry
stands at the intersection of Riemannian geometry and the geometry of
Banach spaces, and our results make it possible to study Banach spaces
in a diﬀerential geometric way.
The article is organized as follows. In Sections 2 and 3, we brieﬂy review the basics of Finsler geometry and optimal transport theory. Then
Section 4 is devoted to Lott, Sturm and Villani’s curvature-dimension
condition (and N -Ricci curvature bound) and its applications. Finally,
we discuss heat ﬂow on Finsler manifolds in Section 5.
Throughout the article, without otherwise indicated, (M, F ) is aconnected, forward complete, n-dimensional C ∞ -Finsler manifold withn ≥ 2, and m is an arbitrary positive C ∞ -measure on M .

§2.

Finsler geometry

We ﬁrst review the basics of Finsler geometry as well as recently
introduced weighted Ricci curvature. We refer to [BCS], [Sh2], [Oh3]and [OS] for further reading.

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Optimal transport and Ricci curvature in Finsler geometry3

2.1. Finsler structures
Let M be a connected, n-dimensional C ∞ -manifold and denote by
π : T M −→ M the natural projection. Given a local coordinate system(xi )ni=1 : U −→ Rn on an open set U ⊂ M , we will always denote by(xi , v i )ni=1 the local coordinate system on π −1 (U ) ⊂ T M given by, forv ∈ π −1 (U ),
n
∑
∂ ¯¯
v=
.
vi i ¯∂x π(v)i=1
Deﬁnition 2.1. (Finsler structures) A C ∞ -Finsler structure of a
C -manifold M is a function F : T M −→ [0, ∞) satisfying the following
conditions:(1) (Regularity) The function F is C ∞ on T M \ 0, where 0 stands forthe zero section.
(2) (Positive homogeneity of degree 1) For any v ∈ T M and positive
number λ > 0, we have F (λv) = λF (v).
(3) (Strong convexity) Given a local coordinate system (xi )ni=1 on U ⊂M , the n × n matrix
( 2 2
)
(
)
1 ∂ (F )(2.1)
gij (v) :=(v)
2 ∂v i ∂v j∞

is positive-deﬁnite at every v ∈ π −1 (U ) \ 0.
In other words, each tangent space (Tx M, F ) is a Minkowski spaceand F varies smoothly in the horizontal direction. We emphasize that,
however, F is not necessarily absolutely homogeneous (or reversible),
namely F (v) 6= F (−v) may happen. It is sometimes helpful to consider←−
the reverse of F , F (v) := F (−v), which turns everything around (e.g.,
distance and geodesics).
For each v ∈ Tx M \ 0, the strong convexity gives the inner product
(2.2)

gv

(∑
n

)
n
n
∑
∂ ∑
∂
ai i ,
bj j :=
ai bj gij (v)∂x j=1 ∂x
i=1
i,j=1

on Tx M . This gv approximates
F in the sense that the unit sphere of√
the norm Fv (w) := gv (w, w) tangents to that of F at v/F (v) up to thesecond order. Note that, if F is coming from a Riemannian structure,then gv always coincides with the original Riemannian structure.
The Legendre transform L∗ : T ∗ M −→ T M associates each covector α ∈ Tx∗ M with a unique vector v = L∗ (α) ∈ Tx M such that
F (v) = F ∗ (α) and α · v = F ∗ (α)2 , where F ∗ stands for the dual

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S. Ohta

Minkowski norm of F on Tx∗ M and α · v is the canonical pairing between Tx M and Tx∗ M . We remark that L∗ |Tx∗ M is linear only if F |Tx M
is an inner product, and L∗ is not diﬀerentiable at the origin. For a C 1 function f : M −→ R, we deﬁne the gradient vector of f at x ∈ M as the
Legendre transform of its diﬀerential, i.e., ∇f (x) := L∗ (Df (x)) ∈ Tx M .
By deﬁnition, ∇f (x) points into the direction in which f increases themost. Observe that
∫ l (
(
)
(
)
(
))
f η(l) − f η(0) ≤F ∇f η(t) dt
0holds for any unit speed C -curve η : [0, l] −→ M . Note the diﬀerence
between ∇(−f ) and −∇f .
For a C 1 -curve η : [0, l] −→ M , we deﬁne its arclength in a natural
way by∫ l()dηL(η) :=F η̇(t) dt,η̇(t) :=
(t).
dt
0
1

Then the corresponding distance function d : M × M −→ [0, ∞) is
given by d(x, y) := inf η L(η), where the inﬁmum is taken over all C 1 curves η from x to y. Unreversibility of F causes nonsymmetry of d,that is, d(x, y) 6= d(y, x). We also remark that the squared distancefunction d(x, ·)2 is C 2 at x if and only if F |Tx M is an inner product([Sh1, Proposition 2.2]). Given x ∈ M and r > 0, we deﬁne the forward
open ball of center x ∈ M and radius r > 0 by
B + (x, r) := {y ∈ M | d(x, y) < r}.
We also deﬁne the open ball in Tx M by
BT+x M (0, r) := {v ∈ Tx M | F (v) < r}.
We say that a C ∞ -curve η : [0, l] −→ M is a geodesic if it is locally minimizing and has constant speed. Deﬁne the exponential map
by expx (v) := η(1) for v ∈ Tx M if there is a geodesic η : [0, 1] −→ Mwith η̇(0) = v. We say that (M, F ) is forward complete if every geodesicη : [0, l] −→ M is extended to a geodesic on [0, ∞), in other words, if
expx is deﬁned on entire Tx M . Then it follows from the Hopf-Rinow
theorem (cf. [BCS, Theorem 6.6.1]) that every pair of two points in M
can be joined by a minimal geodesic.
For each unit vector v ∈ Tx M , let l(v) ∈ (0, ∞] be the supremum ofl > 0 such that the geodesic expx tv is minimal for t ∈ [0, l]. If l(v) < ∞,then the point expx (l(v)v) is called a cut point of x, and the cut locus
Cut(x) of x is deﬁned as the set of all cut points of x.

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Optimal transport and Ricci curvature in Finsler geometry

2.2.

5

Flag and Ricci curvatures

Flag curvature is a substitute of sectional curvature in Riemannian
geometry. We follow the heuristic introduction due to Shen (see [Sh2,
Chapter 6]).
Fix a unit vector v ∈ Tx M and extend it to a C ∞ -vector ﬁeld on an
open neighborhood U of x such that V (x) = v and every integral curve of
V is a geodesic. A typical example is V = ∇[d(η(−ε), ·)] for suﬃciently
small ε > 0, where η : [−ε, ε] −→ M is the geodesic with η̇(0) = v.
The vector ﬁeld V induces the Riemannian structure gV of U through(2.2). Then the ﬂag curvature K(v, w) of v and a linearly independentvector w ∈ Tx M is the sectional curvature with respect to gV of theplane spanned by v and w. (We mean that K(v, w) is independent of
the choice of the vector ﬁeld V .) Similarly, the (Finsler) Ricci curvature
Ric(v) of v is the (Riemannian) Ricci curvature of v with respect to gV .
We remark that the ﬂag curvature K(v, w) depends not only on the ﬂag(the plane spanned by v and w), but also on the ﬂagpole v.
In this setting, for instance, the Bonnet-Myers theorem and theCartan-Hadamard theorem are extended verbatim (cf. [BCS, Theorems
7.7.1, 9.4.1]). We recall several fundamental examples of Finsler manifolds.
Example 2.2. (a) Every Minkowski space (Rn , F ) has ﬂat ﬂag curvature. Here F : Rn −→ [0, ∞) is a positively homogeneous, strongly
convex function and all tangent spaces are canonically identiﬁed with
(Rn , F ).
(b) Unit spheres of Minkowski spaces form an interesting family ofFinsler manifolds, although the author does not know any good reference. There should be a connection with the geometry of Banach spaces.
(c) The Hilbert metric on a bounded, open convex domain D ⊂ Rn
with smooth boundary such that D ∪ ∂D is strictly convex is known toarise from a Finsler metric of constant negative ﬂag curvature.
(d) The Teichmüller metric on Teichmüller space is arguably themost famous Finsler metric in diﬀerential geometry.
(e) If a connected, simply-connected, forward complete Finsler manifold of Berwald type has nonpositive ﬂag curvature, then it is nonpositively curved in the sense of Busemann (due to [KVK, Theorem 7]
combined with the Cartan-Hadamard theorem, see also [KK]). Roughly
speaking, a Finsler manifold of Berwald type is modeled by a single
Minkowski space (all tangent spaces are isometric to each other). Such
Finsler manifolds form a reasonable class including both Riemannian
manifolds and Banach spaces.

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S. Ohta

2.3.

Weighted Ricci curvature

We ﬁx an arbitrary positive C ∞ -measure m on M . Given a unit
vector v ∈ Tx M , recall that v induces the inner product gv on Tx M
through (2.1) and (2.2). Deﬁne the function Ψ on the unit sphere bundle
F −1 (1) ⊂ T M by(
Ψ(v) := log

ωv (BT+x M (0, 1))
mx (BT+x M (0, 1))),

where ωv and mx stands for the Lebesgue measures on Tx M induced
from gv and m, respectively. We can rewrite this as mx = e−Ψ(v) ωv , sothat Ψ is a weight function which lives in F −1 (1), not in M . For brevity,
we set∂v Ψ :=()
d ¯¯¯ Ψ η̇(t) ,dt t=0∂v2 Ψ :=()
d2 ¯¯¯ Ψ η̇(t) ,dt2 t=0

where η : (−ε, ε) −→ M is the geodesic with η̇(0) = v. Inspired by thetheory of weighted Riemannian manifolds ([BE], [Qi], [Lo1]), we haveintroduced the following weighted Ricci curvature in [Oh3].
Deﬁnition 2.3. (Weighted Ricci curvature) Let (M, F ) be an ndimensional C ∞ -Finsler manifold (n ≥ 2) and m be a positive C ∞ measure on M . Given a unit vector v ∈ Tx M , we deﬁne{Ric(v) + ∂v2 Ψ if ∂v Ψ = 0,(i) Ricn (v) :=−∞otherwise,
2
(∂
v Ψ)
(ii) RicN (v) := Ric(v) + ∂v2 Ψ −for N ∈ (n, ∞),N −n2(iii) Ric∞ (v) := Ric(v) + ∂v Ψ.
Note that Ric∞ corresponds to the Bakry-Émery tensor and RicN
is an analogue of Qian’s generalized one. We also remark that bounding
Ricn from below makes sense only when ∂v Ψ = 0 for all v. This is
the case of the Busemann-Hausdorﬀ measure on a Finsler manifold of
Berwald type. However, the existence of such a measure seems a strong
constraint among general Finsler manifolds.
We deﬁne some functions for√ later convenience. For K ∈ R, N ∈(1, ∞) and r ∈ (0, ∞) (r ∈ (0, π (N − 1)/K) if K > 0), we deﬁne




√
√
(N − 1)/K sin(r K/(N − 1))
if K > 0,r
if K = 0,
sK,N (r) :=√ √−(N − 1)/K sinh(r −K/(N − 1)) if K < 0.

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Optimal transport and Ricci curvature in Finsler geometry7

In addition, for t ∈ (0, 1), we deﬁne(tβK,N(r) :=(2.3)

sK,N (tr)tsK,N (r)

)N −1
,

2

2

t
βK,∞
(r) := eK(1−t )r /6 .

Arguing as in weighted Riemannian manifolds (using the Riemannian
structure gV induced from V = ∇d(x, ·)), we immediately obtain a generalized Bishop-Gromov volume comparison theorem.
Theorem 2.4. ([Oh3]) Assume that there are constants K ∈ R andN ∈ [n, ∞) such that RicN √(v) ≥ K holds for all unit vectors v ∈ T M .
Then we have diam √M ≤ π (N − 1)/K if K > 0 and, for any x ∈ M
and 0 < r < R (≤ π (N − 1)/K if K > 0), it holds that∫R
sK,N (t)N −1 dtm(B + (x, R))
0
∫
.
≤
r
m(B + (x, r))s(t)N −1 dt
0 K,NTheorem 2.4 should be compared with Shen’s volume comparison
theorem.
Theorem 2.5. ([Sh1, Theorem 1.1], [Sh2, Theorem 16.1.1]) Assume
that there are constants K ∈ R and H ≥ 0 such that we have Ric(v) ≥ K
and ∂v Ψ ≥ −H for all unit√vectors v ∈ T M . Then we have, for any
x ∈ M and 0 < r < R (≤ π (n − 1)/K if K > 0),∫ R tHe sK,n (t)n−1 dtm(B + (x, R))
0
∫
.
≤
r
m(B + (x, r))
etH sK,n (t)n−1 dt
0Shen’s theorem has a number of topological applications, however,its exponential term etH is troublesome relative to the estimate in Theorem 2.4.
§3.

Optimal transport in Finsler manifolds

In this section, we very brieﬂy recall some basic knowledge of optimaltransport theory for later use. We refer to [Vi1] and recent comprehensive [Vi2] for further reading.
3.1.

The Monge-Kantorovich problem and the Wasserstein distanceLet (M, F ) be a C ∞ -Finsler manifold and denote by P(M ) the setof Borel probability measures on M . For simplicity, we will restrict
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S. Ohta

supported measures. Given µ, ν ∈ Pc (M ), the Monge problem asks how
to ﬁnd and characterize a map T : M −→ M attaining the inﬁmum of
∫
(
)2
(3.1)
d x, T (x) dµ(x)Mamong all maps pushing µ forward to ν (we will write it as T] µ = ν). The
quantity (3.1) can regarded as the cost of transporting µ to ν according
to the map T .
Much later on, Kantorovich extended the Monge problem in a symmetric way. Deﬁne Π(µ, ν) ⊂ P(M ×M ) as the set of couplings of (µ, ν),that is to say, π ∈ Π(µ, ν) if π(A×M ) = µ(A) and π(M ×A) = ν(A) holdfor any measurable set A ⊂ M (in other words, marginals of π are µ andν). Then the Monge-Kantorovich problem is to ﬁnd and characterize an
optimal coupling π ∈ Π(µ, ν) attaining
∫
(3.2)
inf
d(x, y)2 dπ(x, y).
π∈Π(µ,ν)M ×M

The square root of the inﬁmum (3.2) is called the (L2 -)WassersteinW
distance dW
2 (µ, ν) from µ to ν. Note that d2 (µ, ν) is not greater than
the inﬁmum of (3.1) in the Monge problem.
It is known that (Pc (M ), dW2 ) is an Alexandrov space of nonnegative curvature if M is a Riemannian manifold of nonnegative sectional
curvature. This relation can be generalized to Finsler manifolds by introducing the notion of 2-uniform smoothness (see [Oh1], [Oh2], [Sa]).
3.2.

Optimal transport/coupling via (d2 /2)-convex functions

A function ϕ : M −→ R is said to be (d2 /2)-concave if there is afunction ψ : M −→ R such that
ϕ(x) = inf {d2 (x, y)/2 − ψ(y)}y∈Mholds for all x ∈ M . Any (d2 /2)-concave function is Lipschitz continuous and twice diﬀerentiable a.e. ([Oh2, Theorem 7.4]). We say thatϕ is (d2 /2)-convex if −ϕ is (d2 /2)-concave. Then the Brenier-McCanncharacterization of optimal transport states the following. See [Br], [Mc]for original work in Euclidean spaces and Riemannian manifolds.
Theorem 3.1. ([Oh3]) For any µ, ν ∈ Pc (M ) such that µ is absolutely continuous, there exists a (d2 /2)-convex function ϕ : M −→ Rsuch that the map T (x) := expx (∇ϕ(x)) is a unique optimal transportASPM.tex : 2010/1/22 (15:59)

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9from µ to ν in the sense that π := (IdM ×T )] µ is a unique optimal coupling of (µ, ν). Furthermore, the curve (µt )t∈[0,1] given by µt = (Tt )] µwith Tt (x) = expx (t∇ϕ(x)) is a unique minimal geodesic from µ to ν in
(Pc (M ), dW
2 ).
Thus the map T1 is a unique optimal tansport from µ to ν.

§4.

The curvature-dimension condition and its applications

This section is devoted to a main topic of the article, the equivalence between the lower weighted Ricci curvature bound and Lott, Sturmand Villani’s curvature-dimension condition as well as N -Ricci curvaturebound. See [Oh3] for more details.
4.1.

The curvature-dimension condition

In order to deﬁne the N -Ricci curvature bound, we recall the important classes of displacement convex functions (see [LV1], [LV2] fordetails). For N ∈ [1, ∞), denote by DC N 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, ∞). We similarly deﬁne DC ∞ as theset of continuous convex functions U : [0, ∞) −→ R such that U (0) = 0
and that ϕ(s) := es U (e−s ) is convex on R. For an∫ absolutely continuous
measure µ = ρm ∈ P(M ), we deﬁne Um (µ) := M U (ρ) dm. We remarkthat DC N 0 ⊂ DC N if N < N 0 .
The most important element of DC N is U (r)N r(1−r−1/N ) which∫ =1−1/N
derives the Rényi entropy Um (ρm) = N − N M ρ
dm. Letting N
go to inﬁnity provides U (r) = r∫log r ∈ DC ∞ as well as the relative
entropy Entm (ρm) := Um (ρm) = M ρ log ρ dm.
By Theorem 3.1, given two absolutely continuous measures µ0 , µ1 ∈Pc (M ), there is a unique minimal geodesic (µt )t∈[0,1] from µ0 to µ1 in
the Wasserstein space (Pc (M ), dW
2 ). Moreover, every µt is absolutely
continuous (see [Oh3]). Recall (2.3) for the deﬁnition of the functiontβK,N.
Deﬁnition 4.1. (N -Ricci curvature bounds, [LV1], [LV2]) For K ∈R and N ∈ (1, ∞], we say that (M, F, m) has N -Ricci curvature bounded
below by K if, for any two absolutely continuous probability measuresµ0 = ρ0 m, µ1 = ρ1 m ∈ Pc (M ), U ∈ DC N and for any t ∈ (0, 1), it holds

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page: 10

S. Ohtathat)ρ0 (x)Um (µt ) ≤ (1 − t)Udπ(x, y)1−t
ρ0 (x)(d(x, y))
βK,N
M ×M
(
)
∫
t
βK,N
(d(x, y))ρ1 (y)+tUdπ(x, y),t(d(x, y))ρ1 (y)
βK,N
M ×M
∫

1−t
βK,N
(d(x, y))(where (µt )t∈[0,1] is the unique minimal geodesic from µ0 to µ1 and π isthe unique optimal coupling of (µ0 , µ1 ).
In the particular case of K = 0, the above inequality reduces to theconvexity of Um (µ· ). We usually require only the existence of a minimal
geodesic (µt )t∈[0,1] satisfying the above inequality in order to ensure thestability under the (measured Gromov-Hausdorﬀ) convergence of spaces.
Thus we implicitly took advantage of the unique existence of minimal
geodesics in the above deﬁnition.
Sturm’s curvature-dimension condition CD(K, N ) uses the same inequality as Deﬁnition 4.1, but only for the Rényi and the relative entropies ([St2], [St3]). These cases are indeed essential in the sense thatan n-dimensional Riemannian manifold (M, volM ) equipped with the
Riemannian volume element satisﬁes CD(K, N ) if and only if Ric ≥ K
and n ≤ N . The K-convexity of the relative entropy considered in [vRS]
amounts to CD(K, ∞).
Lott, von Renesse, Sturm and Villani’s characterization of weighted
Ricci curvature bounds ([vRS], [St1], [St2], [St3], [LV1], [LV2]) is successfully generalized to our Finsler setting.
Theorem 4.2. ([Oh3]) For K ∈ R and N ∈ [n, ∞], the following
conditions are equivalent:(i) RicN (v) ≥ K holds for any unit vector v ∈ T M .
(ii) (M, F, m) satisﬁes the curvature-dimension condition CD(K, N ).
(iii) (M, F, m) has N -Ricci curvature bounded below by K.
The proof of the above theorem heavily depends on careful analysis
of optimal transport maps developed in [CMS]. Roughly speaking, as an
optimal transport Tt goes along the gradient vector ﬁeld of some (d2 /2)convex function (Theorem 3.1) which is twice diﬀerentiable a.e., we candiﬀerentiate Tt in the space variable. Then we obtain Jacobi ﬁelds andit is natural to control their behavior using the Ricci curvature.
As was brieﬂy mentioned, the N -Ricci curvature bound as well as
the curvature-dimension condition are preserved under the measured
Gromov-Hausdorﬀ convergence ([St2], [St3], [LV1], [LV2]). Therefore the
N -Ricci curvature bound is useful in the investigation of limit spaces ofASPM.tex : 2010/1/22 (15:59)

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Optimal transport and Ricci curvature in Finsler geometry

11

Finsler manifolds with uniform lower (weighted) Ricci curvature bounds.
We refer to [CC] for related, celebrated work in the Riemannian case,and remark that even Banach spaces can not be approximated by Riemannian spaces with a uniform lower Ricci curvature bound.
4.2.

Interpolation inequalities

The technique used in the implication (i) ⇒ (ii) of Theorem 4.2
has been established in Cordero-Erausquin, McCann and Schmuckenschläger’s remarkable work [CMS]. They used it to study several interpolation inequalities on Riemannian manifolds. We refer to [Ga], [Le]for basics and further reading in the Euclidean setting.
The classical Brunn-Minkowski inequality in the Euclidean space
Rn asserts the concavity of the function t 7−→ voln ((1 − t)A + tB)1/n
of t ∈ [0, 1] for any (nonempty) bounded, measurable sets A, B ⊂ Rn ,where voln stands for the Lebesgue measure and(1 − t)A + tB := {(1 − t)x + ty | x ∈ A, y ∈ B}.
The following nonlinear version is introduced in [St3] (see also [vRS] forthe inﬁnite dimensional case) and used to establish the implication (ii)
⇒ (i) of Theorem 4.2 in the Riemannian case. Given subsets A, B ∈ M
and t ∈ (0, 1), we denote by Zt (A, B) the set of points η(t) such that
η : [0, 1] −→ M is a minimal geodesic with η(0) ∈ A and η(1) ∈ B.
Theorem 4.3. (Brunn-Minkowski inequality, [St3], [Oh3]) Supposethat there are constants K ∈ R and N ∈ [n, ∞] such that RicN (v) ≥ K
holds for all unit vectors v ∈ T M .
(i)

If N < ∞, for any nonempty measurable sets A, B ⊂ M andt ∈ (0, 1), we have(
)1/N
m Zt (A, B)≥ (1 − t)
+t

(ii)

inf

(
)1/N
1−t
βK,N
d(x, y)· m(A)1/N

inf

(
)1/N
t
βK,N
d(x, y)· m(B)1/N .

x∈A,y∈Bx∈A,y∈B

If N = ∞, for any bounded measurable sets A, B ⊂ M with
m(A), m(B) > 0 and t ∈ (0, 1), we have()
log m Zt (A, B) ≥ (1 − t) log m(A) + t log m(B)K2
+ (1 − t)tdW
2 (µ, ν) ,2
where µ = m(A)−1 m|A and ν = m(B)−1 m|B .

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S. Ohta

The classical Brunn-Minkowski inequality in Rn admits a functionalversion called the Prékopa-Leindler inequality. It is also extended to thenonlinear setting.
Theorem 4.4. (Prékopa-Leindler inequality, [CMS], [Oh3]) Suppose that there are constants K ∈ R and N ∈ [n, ∞] such that RicN (v) ≥K holds for all unit vectors v ∈ T M . Take t ∈ (0, 1), three nonnegative
measurable functions∫ f, g, h : M∫ −→ [0, ∞) ∫and nonempty
measurable∫
sets A, B ⊂ M with A f dm = M f dm and B g dm = M g dm. If(h(z) ≥f (x)1−tβK,N (d(x, y)))1−t (g(y)
tβK,N(d(x, y))

)t

holds
for all x∫∈ A, y ∈ B and z ∈ Zt (x, y), then we have∫
( M f dm)1−t ( M g dm)t .

∫M

h dm ≥

The above Prékopa-Leindler inequality (with N < ∞) is the special
case (p = 0) of the p-mean inequality studied in [CMS]. The following (Borell-)Brascamp-Lieb inequality is also the special as well as thestrongest case (p = −1/N ) of the p-mean inequality. See [CMS], [Ga]for more details.
Theorem 4.5. (Brascamp-Lieb inequality, [CMS], [Oh3]) Assume
that there are constants K ∈ R and N ∈ [n, ∞) such that RicN (v) ≥ K
holds for all unit vectors v ∈ T M . Take three nonnegative measurable∫functions f,∫ g, h : M −→ [0, ∞) and measurable sets A, B ⊂ M withf dm = B g dm = 1. If there is t ∈ (0, 1) such that
A
( 1−t
)
( t
)
βK,N (d(x, y)) 1/N
βK,N (d(x, y)) 1/N
1≤ (1 − t)
+t
f (x)
g(y)
h(z)1/N∫holds for all x ∈ A, y ∈ B and z ∈ Zt (x, y), then we have M h dm ≥ 1.
4.3. Functional inequalities and concentration of measures
Connection between the condition like N -Ricci curvature bound andseveral functional inequalities is ﬁrst indicated in Otto and Villani’s
inﬂuencial paper [OV], and then systematically investigated by Lott andVillani [LV1], [LV2]. We ﬁrst recall results in the inﬁnite dimensional
case.
Theorem 4.6. ([LV1], [Oh3]) Let (M, F, m) be a compact Finsler
manifold satisfying m(M ) = 1 and Ric∞ (v) ≥ K for some K ∈ R and
all unit vectors v ∈ T M . Then the following hold.

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Optimal transport and Ricci curvature in Finsler geometry(i)(Talagrand inequality/Transport cost inequality) For any absolutely continuous measure µ ∈ P(M ), we have
Entm (µ) ≥

(ii)

13

{
}
K
2 W
2
max dW
2 (m, µ) , d2 (µ, m) .
2

(Logarithmic Sobolev inequality)
For any Lipschitz continuous∫function f : M −→ R with M f 2 dm = 1, we have
(∫
)1/2
K W
2
Entm (µ) ≤ 2
F (∇|f |) dm
dWd (m, µ)2 ,
2 (m, µ) −2 2
Mwhere we set µ = f 2 m. In particular, if K > 0, then it holds that∫2
Entm (µ) ≤F (∇f )2 dm.
K M

(iii)

(Global Poincaré inequality) If K > 0, then we∫ have, for any
Lipschitz continuous function f : M −→ R with M f dm = 0,∫∫1
f 2 dm ≤F (∇f )2 dm.
KM
M

The above global Poincaré inequality can be sharpended in the ﬁnite
dimensional case as follows.
Theorem 4.7. (Lichnerowicz inequality, [LV2], [Oh3]) Suppose that
there are constants K > 0 and N ∈ [n, ∞) such that RicN (v) ≥ K holds
for all unit vectors v ∈ T M .∫ Then we have, for any Lipschitz continuousfunction f : M −→ R with M f dm = 0,∫∫
N −1
f 2 dm ≤F (∇f )2 dm.
KN M
MInterpolation and functional inequalities are closely related to theconcentration of measure phenomenon (see [Le]). For instance, we cangeneralize Gromov and Milman’s Gaussian (normal) concentration result of Riemannian manifolds of positive Ricci curvature ([GM]). For a(symmetric or nonsymmetric) metric space (X, d) equipped with a Borel
probability measure µ on X, we deﬁne the concentration function by{()}α(X,d,µ) (r) := sup 1 − µ B + (A, r) | A ⊂ X, µ(A) ≥ 1/2
for r > 0. Here we set B + (A, r) := {y ∈ X | inf x∈A d(x, y) < r}.
Proposition 4.8. ([Oh3]) If there is a positive constant K > 0such that Ric∞ (v) ≥ K holds for all unit vectors v ∈ T M , then we have2α(M,F,m) (r) ≤ 2e−Kr /4 for all r > 0.

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14

§5.

S. Ohta

Laplacian and the heat equation

In this ﬁnal section, we brieﬂy review the contents of [OS]. For
simplicity, we will treat only compact Finsler manifolds.
5.1.

Finsler Laplacian

For a vector ﬁeld V on M diﬀerentiable a.e., we deﬁne its divergence
div V : M −→ R through the identity∫∫∫
φ div V dm = −V φ dm = −Dφ · V dm
M

M

M

for all φ ∈ C ∞ (M ), where Dφ(x) · V (x) denotes the canonical paring
between Tx M and Tx∗ M . We remark that div V depends on the choiceof the measure m. If V is diﬀerentiable in a local coordinate system(xi )ni=1 , then we have the explicit formuladiv V (x) =

n {
∑
∂V i
i=1

}
∂Ψ
i
(x)
−
V
(x)
(x)
,
∂xi
∂xi

∑n

where V = i=1 V i (∂/∂xi ) and m(dx) = e−Ψ(x) dx1 · · · dxn in this coordinate. The energy functional E : H 1 (M ) −→ [0, ∞) is deﬁned by∫∫
1
1F ∗ (Du)2 dm =F (∇u)2 dm.
E(u) :=2 M
2 M
For later use, we also set{∫¯¯
λ := inf 2E(u) ¯ u ∈ H 1 (M ),}u dm = 0, kukL2 = 1 .

M

It is immediate by virtue of the compactness of M that λ > 0. Using
the inner product gv in (2.2), we also introduce the constant κ ∈ (0, 1]
by
F (w)2
κ := inf
inf.
x∈M v,w∈F −1 (1)∩Tx M gv (w, w)√
Note that 1/ κ coincides with the 2-uniform smoothness constant studied in [Oh2], and that κ = 1 holds if and only if (M, F ) is Riemannian.
We observe by deﬁnition
that the functional E is (κλ)-convex on the set∫{u ∈ L2 (M ; m) | M u dm = C} for each constant C ∈ R.
Deﬁnition 5.1. (Finsler Laplacian) We deﬁne the Finlser Laplacian ∆ acting on functions u ∈ H 1 (M ) formally as ∆u := div(∇u).

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Optimal transport and Ricci curvature in Finsler geometry

15

To be precise, ∆u is the distributional Laplacian deﬁned through theidentity∫∫φ∆u dm = −M

Dφ · ∇u dm
M

for all φ ∈ C ∞ (M ).
Recall that, even when u is C ∞ , the gradient vector ﬁeld ∇u is not
diﬀerentiable at points x with ∇u(x) = 0. Our Finsler Laplacian is anonlinear operator because the Legendre transform is not∫ linear. Notealso that our Laplacian is a negative operator, that is, M u∆u dm ≤ 0
holds for all u ∈ H 1 (M ). The same idea as Theorem 2.4 (using gV with
V = ∇d(x, ·)) leads us to the Laplacian comparison theorem.
Theorem 5.2. (Laplacian comparison, [OS]) Assume that there areconstants K ∈ R and N ∈ [n, ∞) such that RicN (v) ≥ K holds for
all unit vectors v ∈ T M . Then the Laplacian of the distance functionu = d(x, ·) from any ﬁxed point x ∈ M satisﬁes∆u(z) ≤ (N − 1)

s0K,N (d(x, z))sK,N (d(x, z))pointwise on M \ ({x} ∪ Cut(x)) and in the sense of distribution on
M \ {x}.
5.2. The heat equation
We formulate the heat equation associated with our Finsler Laplacian and show the unique existence of its solution as gradient ﬂow of theenergy functional on L2 (M ; m).
Deﬁnition 5.3. We say that a function u is a (global) solution to
the heat equation ∂t u = ∆u on [0, T ] × M if u ∈ L2 ([0, T ], H 1 (M )) ∩H 1 ([0, T ], H −1 (M )) and if, for every t ∈ [0, T ] and φ ∈ C ∞ (M ), it holdsthat∫∫φ∂t ut dm = −Dφ · ∇ut dm,
MM

where ut (x) := u(t, x).
The compactness of M ensures∫ that every ∫solution u to the heat
equation is mass preserving, i.e., M ut dm = M u0 dm holds for allt > 0.
We can construct a solution to the heat equation starting from anarbitrary initial point u0 ∈ H 1 (M ) as gradient ﬂow of the energy functional E on L2 (M ; m). As L2 (M ; m) is a Hilbert space, we can certainly
apply Crandall and Liggett’s classical technique ([CL], see also [AGS]).

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S. Ohta

Theorem 5.4. ([OS]) For each u0 ∈ H 1 (M ) and T > 0, there exists a unique solution u to the heat equation lying in L2 ([0, T ], H 1 (M )) ∩H 1 ([0, T ], L2 (M )). Moreover, for each t ∈ (0, T ), the distributional
Laplacian ∆ut is absolutely continuous with respect to m and we havelimδ↓0E(ut ) − E(ut+δ )= k∆ut k2L2
δfor all t > 0.
Uniqueness in Theorem 5.4 is a consequence of the L2 -contraction
property
that, given two solutions u, v to the heat equation∫ which asserts∫with M u dm = M v dm,
kut − vt kL2 ≤ e−tκλ ku0 − v0 kL2
holds for all t > 0. Note that the L2 -contraction property follows fromthe (κλ)-convexity of E, for a solution to the heat equation is gradient
ﬂow of E in L2 (M ; m).
As for regularity, we can show that every solution to the heat equation admits a Hölder continuous version and it is H 2 in the space variablex as well as C 1,α in both variables t and x for some α > 0. However, u
is not C 2 in general. The typical (noncompact) example is a Gaussian
kernel of a Minkowski space, which is C 2 at the origin if and only if the
Minkowski space happens to be a Hilbert space.
5.3. The Finsler structure of the Wasserstein space
We next present another characterization of heat ﬂow as gradient
ﬂow of the relative entropy in the reverse Wasserstein space (i.e., theWasserstein space with respect to the reverse Finsler structure). To doso, we introduce a Finsler structure of the Wasserstein space in accordance with Otto’s formulation in [Ot] (see also [AGS], [Lo2]).
Given arbitrary ϕ ∈ C 2 (M ), the function εϕ is (d2 /2)-convex for
suﬃciently small ε > 0. This observation (together with Theorem 3.1)
leads us to the following notion of tangent space:
Tµ P := {V = ∇ϕ | ϕ ∈ C ∞ (M )},where the closure is taken with respect to the Minkowski norm
)1/2
(∫
(
)2
FW (µ, V ) :=F V (x) dµ(x).
M

Note the dependence of FW (µ, ·) on the base point µ, which generates
the nontrivial geometric structure of (P(M ), FW ).

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Optimal transport and Ricci curvature in Finsler geometry

17

Every locally Lipschitz continuous curve (µt )t∈(0,1) is accompaniedwith the unique Borel vector ﬁeld Ψt (x) = Ψ(t, x) ∈ Tx M on (0, 1) × M
with F (Ψ) ∈ L∞
loc ((0, 1) × M ; dµt dt) satisfying the continuity equation∂t µt + div(Ψt µt ) = 0
in the weak sense that∫ 1∫{∂t φt + Dφt · Ψt } dµt dt = 00Mholds for all φ ∈ Cc∞ ((0, 1) × M ). We consider Ψ as the tangent vector
ﬁeld of the curve (µt )t∈(0,1) and denote it by µ̇t = Ψt ∈ Tµt P.
Given µ, ν ∈ P(M ), we have(∫ 1dW
2 (µ, ν) =inf(µt )t∈[0,1])1/2FW (µt , µ̇t ) dt,

0

where (µt )t∈[0,1] runs over all Lipschitz continuous curves in P(M ) with
µ0 = µ and µ1 = ν. Thus our Finsler structure realizes the Wasserstein
distance.
5.4. Heat ﬂow as gradient ﬂow in the Wasserstein space
We deﬁne the cotangent space at given µ ∈ P(M ) in a similar manner to the tangent space as
Tµ∗ P := {α = Dϕ | ϕ ∈ C ∞ (M )},
(∫
)1/2
(
)2
∗
∗
FW (µ, α) :=F α(x) dµ(x),M∗where the closure is taken with respect to FW(µ, ·). Then the associatedLegendre transform L∗W (µ, ·) : Tµ∗ P −→ Tµ P at µ is deﬁned in the pointwise way by L∗W (µ, α)(x) := L∗ (α(x)). We also deﬁne the exponential
map expµ : Tµ P −→ P(M ) by expµ (V ) := (exp V )] µ.
Given a function S on (a subset of) P(M ), we say that S is diﬀerentiable at µ ∈ P(M ) if there is α ∈ Tµ∗ P such that∫

S(expµ (tV )) − S(µ)
t↓0tα · V dm = limMholds for all V = ∇ϕ ∈ Tµ P with ϕ ∈ C ∞ (M ). Then we denote α byDS(µ) and call it the derivative of S at µ, and the gradient vector of S
at µ is deﬁned as ∇W S(µ) := L∗W (µ, DS(µ)).

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S. Ohta

Deﬁnition 5.5. (Gradient ﬂow) A continuous curve (µt )t≥0 ⊂P(M ) which is locally Lipschitz continuous on (0, ∞) is called a gradient ﬂow of S if µ̇t = ∇W (−S)(µt ) holds at a.e. t ∈ (0, ∞).
Note the diﬀerence between ∇W (−S) and −∇W S. The gradientvector of − Entm can be described explicitly.
Proposition 5.6. ([OS]) Take absolutely continuous µ = ρm ∈P(M ) with ρ ∈ H 1 (M ). If − log ρ 6∈ H 1 (M ; µ), then − Entm is not
diﬀerentiable at µ. If − log ρ ∈ H 1 (M ; µ), then − Entm is diﬀerentiable
at µ and the gradient vector is given by∇W (− Entm )(µ) =1∇(−ρ) ∈ Tµ P.
ρIn particular, its norm squared FW (µ, ∇W (− Entm )(µ))2 coincides with←−
the Fisher information with respect to the reverse Finsler structure F :
∫
∫
(
)2 1
←
−
←
−(←
− )2 1
I (µ) :=F ∇ρdm =F ∇(−ρ)dm.
ρρMM
We ﬁnally obtain a characterization of heat ﬂow as gradient ﬂow of
the relative entropy in the reverse Wasserstein space. Compare this with
the Euclidean and Riemannian cases in [JKO], [Oh1] and [Sa].
Theorem 5.7. ([OS]) Let (µt )t≥0 ⊂ P(M ) be a continuous curve
which is locally Lipschitz continuous on (0, ∞), and assume that µt =
ρt m with ρt ∈ H 1 (M ) a.e. t ∈ (0, ∞). Then (µt )t≥0 is a gradient ﬂow
of the relative entropy if and only if (ρt )t≥0 is a heat ﬂow with respect←−
to the reverse Finsler structure F of F .

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Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 6068502, JAPANE-mail address: sohta@math.kyoto-u.ac.jp