Nonlinear geometric analysis on Finsler manifolds
Shin-ichi Ohta∗

Abstract
This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our
purpose is two-fold: Give a concise and geometric review on the birth of weighted
Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the Γ-calculus based on the Bochner inequality. In the latter we discuss
some gradient estimates, functional inequalities, and isoperimetric inequalities.

Contents
1 Introduction

2

2 Comparison geometry of Finsler manifolds
4
2.1 Finsler structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Asymmetric distance and geodesics . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Weighted Ricci curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Nonlinear Laplacian and heat semigroup
3.1 Gradient vectors, Hessian and Laplacian . . . . . . . . . . . . . . . . . . .
3.2 Bochner–Weitzenböck formula . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
15
16

4 Linearized heat semigroups and gradient estimates
4.1 Linearized heat semigroups and their adjoints . . . . . . . . . . . . . . . .
4.2 Gradient estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Characterizations of lower Ricci curvature bounds . . . . . . . . . . . . . .

17
18
19
21

5 Functional inequalities
5.1 Poincaré–Lichnerowicz inequality . . . . . . . . . . . . . . . . . . . . . . .
5.2 Logarithmic Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21
22
23
25

∗

Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan (sohta@math.kyotou.ac.jp), Current address: Department of Mathematics, Osaka University, Osaka, 560-0043, Japan(s.ohta@math.sci.osaka-u.ac.jp); Supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K04844.

1

6 Gaussian isoperimetric inequality28
6.1 Background for Lévy–Gromov isoperimetic inequality . . . . . . . . . . . . 29
6.2 Isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1Introduction

The aim of this article is to review the recent developments of comparison geometry
and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below.
The weighted Ricci curvature was introduced by the author in [Oh3]. Since then it hashelped us to understand the similarities and diﬀerences between Riemannian and Finsler
manifolds more deeply. One of the main challenges, compared with the Riemannian case,is in the nonlinearity of the Laplacian and the associated heat equation. How to deal withsuch a nonlinear evolution equation would be of interest also in the analytic viewpoint.
A Finsler manifold will be a pair of a manifold M and a nonnegative function F on thetangent bundle T M such that F |Tx M gives a Minkowski norm for each x ∈ M . We remarkthat F (−v) ̸= F (v) is allowed, which is called the non-reversibility and is a special featureof Finsler manifolds. One can define the distance and geodesics in natural geometric ways,
whereas the non-reversibility of F leads to the asymmetry of the distance, namely d(y, x)does not necessarily coincide with d(x, y). Analyzing the behavior of geodesics, we canfurther introduce Jacobi fields and the curvature tensor. Thus we arrive at the natural
notions of the flag curvature (a generalization of the Riemannian sectional curvature) and
the Ricci curvature. Some results in comparison Riemannian geometry concerning the
distance are then generalized to the Finsler context (for example, the Cartan–Hadamard
theorem and the Bonnet–Myers theorem).
Now, to proceed further in this direction, what is a measure controlled by the Ricci curvature? At this point we have another diﬃculty; There are several choices for a ‘canonical’
measure (such as the Busemann–Hausdorﬀ measure and the Holmes–Thompson measure),
all of them go down to the volume measure on a Riemannian manifold. Our strategy initiated in [Oh3] is that, instead of choosing some constructive measure, we begin withan arbitrary (smooth, positive) measure m on M and modify the Ricci curvature according to the choice of m. The modification is done in the same manner as weighted
Riemannian manifolds. We call this modified Ricci curvature the weighted Ricci curvature RicN (also called the Bakry–Émery–Ricci curvature), where N is a parametersometimes called the eﬀective dimension. By means of RicN , one can generalize many results including the Bishop–Gromov volume comparison to this Finsler context and, mostnotably, the curvature-dimension condition CD(K, N ) in the sense of Lott–Sturm–Villani([St1, St2, LV]) is equivalent to the lower curvarture bound RicN ≥ K ([Oh3]). This
deep equivalence relation ensures the naturalness and importance of the weighted Ricci
curvature.
The Ricci curvature plays the prominent role in geometric analysis. The Laplacian ∆acting on functions is naturally defined as the divergence (in terms of m) of the gradientvector field (in terms of F ). This is also called the Witten Laplacian or the weighted
Laplacian. Since taking the gradient vector (more precisely, the Legendre transform) is
a nonlinear operation, our Laplacian is nonlinear (unless F comes from a Riemannian2

metric). Nonetheless, since the Laplacian is (locally) uniformly elliptic, the associatedheat equation ∂t u = ∆u is still well-posed. In fact, we can apply the classical technique
due to Saloﬀ-Coste [Sal] and others to show the unique existence and a certain regularityof solutions to the heat equation ([GS, OS1]).
Furthermore, the Bochner–Weitzenböck formula was established in [OS3] in the formof
[ 2]
∇u F (∇u)∆− D[∆u](∇u) = Ric∞ (∇u) + ∥∇2 u∥2HS(∇u) ,2
followed by the Bochner inequality
]
[ 2
(∆u)2
∇u F (∇u)∆− D[∆u](∇u) ≥ KF 2 (∇u) +2Nunder the lower curvature bound RicN ≥ K. Here we mixed the nonlinear Laplacian∆ and its linearization ∆∇u . Similarly to the Riemannian case, the Bochner inequality
plays a significant role. See [OS3, Oh8, WX, Xi, YH] for some geometric and analytic
applications not covered in this article.
Our focus in this article will be on a nonlinear analogue of the Γ-calculus. The Γcalculus, developed by Bakry and his collaborators (see [BE, Bak] and the recent book
[BGL]), is a successful theory of analyzing linear diﬀusion operators and the associatedsemigroups by means of the integration by parts and a kind of Bochner inequality. This
theory fits surprisingly well with our Finsler setting, although the Laplacian is nonlinear.
We will discuss three applications of the Γ-calculus: gradient estimates, functional
inequalities, and isoperimetric inequalities. We will give the outlines of some proofs toshow the basic idea of the Γ-calculus. We first consider the L2 - and L1 -gradient estimatesfor heat flow under Ric∞ ≥ K. These gradient estimates are indeed equivalent to Ric∞ ≥K similarly to the Riemannian situation ([vRS, Oh11]). Next we show three functional
inequalities: the Poincaré–Lichnerowicz (spectral gap) inequality, the logarithmic Sobolev
inequality, and the Sobolev inequality. Here the assumption is RicN ≥ K > 0 withN ∈ [n, ∞], and N can be also negative only in the Poincaré–Lichnerowicz inequality([Oh12]). We finally study the Gaussian isoperimetric inequality under Ric∞ ≥ K > 0,
which generalizes Bakry–Ledoux’s inequality in the Riemannian situation ([BL, Oh11]).
All the estimates are sharp and have the same forms as Riemannian manifolds. Only
the exception is the admissible range of the exponent p in the Sobolev inequality (seeRemark 5.12). We stress that, however, we cannot generalize every result of Riemannian
manifolds to Finsler manifolds. One of the most important diﬀerences is the lack ofcontraction property of heat flow ([OS2]). In general we know much less about gradient
flows of convex functions in Finsler manifolds or even in normed spaces (see [OP] for a
related discussion).
The organization of the article is as follows. Section 2 is a crash course for the basic
notions and ideas of comparison Finsler geometry. We try to explain in a geometric way
how the distance, geodesics, and curvatures arise. Section 3 is devoted to the study of the
Laplacian and the heat semigroup. Then we consider the gradient estimates in Section 4,
the functional inequalities in Section 5, and the isoperimetric inequality in Section 6,
respectively.
3

Acknowledgements. I thank Professor Sorin Sabau for encouraging me to write a survey
on this subject. I am also grateful to an anonymous referee for valuable comments.

2

Comparison geometry of Finsler manifolds

This section is devoted to a concise review on comparison geometry of Finsler manifolds.
The main object will be a Finsler manifold equipped with a measure whose weighted Ricci
curvature is bounded from below. We refer to [BCS, Sh2, SS] for the fundamentals of
Finsler geometry as well as related comparison geometric studies.
Throughout the article, unless otherwise indicated, let M be a connected C ∞ -manifold
without boundary of dimension n ≥ 2. We will also fix an arbitrary positive C ∞ -measurem on M from Subsection 2.5.

2.1

Finsler structures

Given local coordinates (xi )ni=1 on an open set U ⊂ M , we will always use the fiber-wiselinear coordinates (xi , v j )ni,j=1 of T U such that
v=

n
∑
j=1

vj

∂
∈ Tx M,∂xj x

x ∈ U.

Definition 2.1 (Finsler structures) We say that a nonnegative function F : T M −→[0, ∞) is a C ∞ -Finsler structure of M if the following three conditions hold:(1) (Regularity) F is C ∞ on T M \ 0, where 0 stands for the zero section;(2) (Positive 1-homogeneity) It holds F (cv) = cF (v) for all v ∈ T M and c ≥ 0;(3) (Strong convexity) The n × n matrix
(

)n

(

gij (v) i,j=1 :=)n
1 ∂ 2 (F 2 )(v)
2 ∂v i ∂v j
i,j=1

(2.1)

is positive-definite for all v ∈ T M \ 0.
We call such a pair (M, F ) a C ∞ -Finsler manifold.
Notice that the positive-definiteness is independent of the choice of local coordinates.
One can similarly define C l -Finsler manifolds, though we consider only C ∞ -Finsler manifolds in this article. Some more remarks on Definition 2.1 are ready.
Remark 2.2 (a) The homogeneity (2) is imposed only in the positive direction, that is,for nonnegative c. This leads to the asymmetry of the associated distance function (seethe next subsection). If F (−v) = F (v) holds for all v ∈ T M , then we say that F isreversible or absolutely homogeneous.
(b) The strong convexity (3) means that the unit sphere Ux M := Tx M ∩ F −1 (1)(called the indicatrix ) is positively curved, and it implies the strict convexity: F (v + w) ≤4

F (v) + F (w) for all v, w ∈ Tx M and equality holds only when v = aw or w = av for somea ≥ 0.
(c) Although we will discuss only under Definition 2.1, the C ∞ -regularity (1) and thestrong convexity (3) can be weakened in various ways occasionally (see [OS1] for instance).
Let us continue the study of the strong convexity in the remainder of this subsection.
Given each v ∈ Tx M \ 0, the positive-definite matrix (gij (v))ni,j=1 in (2.1) induces the
Riemannian structure gv of Tx M as
gv

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

(∑
n

(2.2)

Notice that this definition is coordinate-free, and we have gv (v, v) = F 2 (v). One canregard gv as the best Riemannian approximation of F |Tx M in the direction v. In fact,
the unit sphere of F |Tx M (which is positively curved due to Remark 2.2(b)) is tangent tothat of gv at v/F (v) up to the second order. The metric gv plays quite important roles incomparison Finsler geometry.
∑In the coordinates (xi , αj )ni,j=1 of T ∗ U given by α = nj=1 αj dxj , we will also consider
gij∗ (α) :=1 ∂ 2 [(F ∗ )2 ](α),
2 ∂αi ∂αji, j = 1, 2, . . . , n,for α ∈ T ∗ U \ 0. Here F ∗ : T ∗ M −→ [0, ∞) is the dual Minkowski norm to F , namelyF ∗ (α) :=supα(v) =v∈Tx M, F (v)≤1

sup

α(v)

v∈Tx M, F (v)=1for α ∈ Tx∗ M . It is clear by definition that α(v) ≤ F ∗ (α)F (v), and henceα(v) ≥ −F ∗ (α)F (−v),α(v) ≥ −F ∗ (−α)F (v).

We remark and stress that, however, α(v) ≥ −F ∗ (α)F (v) does not hold in general due tothe non-reversibility of F .
The strong convexity is related to the following quantities, these are fundamental in
the geometry of Banach spaces (see [BCL, Oh2]). For x ∈ M , we define the (2-)uniform
smoothness constant at x by
SF (x) :=gv (w, w)
F ∗ (β)2
.
=
sup
2
∗
v,w∈Tx M \0 F (w)α,β∈Tx∗ M \0 gα (β, β)
supSince gv (w, w) ≤ SF (x)F 2 (w) and gv is the Hessian of F 2 /2 at v, the constant SF (x)measures the concavity of F 2 in Tx M . We also set SF := supx∈M SF (x). Notice thatSF ∈ [1, ∞] and SF = 1 holds if and only if F comes from a Riemannian metric (see[Oh2]). We similarly define the (2-)uniform convexity constants as
F 2 (w)gα∗ (β, β)
CF (x) := sup
= sup
,
∗
2
v,w∈Tx M \0 gv (w, w)
α,β∈Tx∗ M \0 F (β)
5CF := sup CF (x).
x∈M

Again, CF ∈ [1, ∞] in general and CF = 1 holds if and only if (M, F ) is Riemannian.
It is readily seen that the constants SF and CF control the reversibility constant,defined by
F (v)ΛF := sup∈ [1, ∞],v∈T M \0 F (−v)
as follows (see [Oh11, Lemma 2.4] for instance).
Lemma 2.3 We have√√ΛF ≤ min{ SF , CF }.

For later convenience, we introduce the following notations.
Definition 2.4 (Reverse Finsler structures) We define the reverse Finsler structure←−←−
F of F by F (v) := F (−v). We will put an arrow ← on those quantities associated with
←
−
F.
←
−
←−
←−
For example, d(x, y) = d(y, x) (see §2.2), Ric(v) = Ric(−v) (§2.4), RicN (v) =←−←−RicN (−v) (§2.5), ∇u = −∇(−u) and ∆u = −∆(−u) (§3.1).

2.2

Asymmetric distance and geodesics

For x, y ∈ M , we define the (asymmetric) distance from x to y by
∫ 1()
d(x, y) := inf
F η̇(t) dt,
η0

where the infimum is taken over all piecewise C 1 -curves η : [0, 1] −→ M such that η(0) = x
and η(1) = y. Note that the asymmetry d(y, x) ̸= d(x, y) can occur since F is only
positively homogeneous (so d is not properly a distance function in the usual sense). We
also remark that the squared distance function d2 (x, ·) is only C 1 at x in general, and thatd2 (x, ·) is C 2 at x for all x ∈ M if and only if F comes from a Riemannian metric (see[Sh1, Proposition 2.2]). This is a reason why we have the less regularity for heat flow (seeTheorem 3.7 below).
In the manner of metric geometry, we say that a C ∞ -curve η : I −→ M from aninterval I ⊂ R is geodesic if it is locally minimizing and has a constant speed with respect
to d. Precisely, there is C ≥ 0 and, for any t ∈ I, we find ε > 0 such that()
d η(s), η(s′ ) = C(s′ − s)for all s, s′ ∈ I ∩ [t − ε, t + ε] with s ≤ s′ (then F (η̇) ≡ C as a matter of course).
One can write down the geodesic equation as the Euler–Lagrange equation for the action induced from F 2 . In such calculations, the following basic theorem plays fundamentalroles (see [BCS, Theorem 1.2.1]).

6

Theorem 2.5 (Euler’s homogeneous function theorem) Consider a diﬀerentiable
function H : Rn \ {0} −→ R satisfying H(cv) = cr H(v) for some r ∈ R and all c > 0 andv ∈ Rn \ {0} (that is, H is positively r-homogeneous). Then we have
n
∑
∂H

∂v i
i=1

(v)v i = rH(v)for all v ∈ Rn \ {0}.
Observe that gij defined in (2.1) is positively 0-homogeneous on each Tx M , and hencen∑i

Aijk (v)v =

i=1

n
∑

j

Aijk (v)v =

j=1

n
∑

Aijk (v)v k = 0

(2.3)

k=1

for all v ∈ T M \ 0 and i, j, k = 1, 2, . . . , n, where
Aijk (v) :=F (v) ∂gij(v),
2 ∂v kv ∈ T M \ 0,is the Cartan tensor which measures the variation of gv in the vertical directions. The
Cartan tensor vanishes everywhere on T M \ 0 if and only if F comes from a Riemannian
metric. In this sense, the Cartan tensor is a genuinely non-Riemannian quantity.
With the help of (2.3), we arrive at the following geodesic equation by the usual
calculation similar to the Riemannian case:η̈ i (t) +

n
∑

(
)
i
γjk
η̇(t) η̇ j (t)η̇ k (t) = 0

(2.4)

j,k=1

for all i, where
{
}
∂glk
∂gjl
∂gjk
g (v)
(v) + k (v) −
(v) ,
2 l=1
∂xj
∂x
∂xl

1
i
γjk
(v) :=n∑ilv ∈ T M \ 0,(2.5)

is called the formal Christoﬀel symbol. We denoted by (g ij (v)) the inverse matrix ofi(gij (v)). Notice that γjk
has the same form as the Riemannian Christoﬀel symbol, while
it is a (0-homogeneous) function on T M \ 0 and cannot be reduced to a function on M .
By the general ODE theory, every initial vector v ∈ T M admits a unique geodesicη : (−ε, ε) −→ M with η̇(0) = v for some ε > 0. Given v ∈ Tx M , if there is a geodesicη : [0, 1] −→ M with η̇(0) = v, then we define the exponential map by expx (v) := η(1).
We say that (M, F ) is forward complete if the exponential map is defined on whole T M .
In other words, every geodesic η : [0, 1] −→ M is extended infinitely in the forward
direction to the geodesic η̄ : [0, ∞) −→ M . If every geodesic η : [0, 1] −→ M is extended←−
in the backward direction to the geodesic η̄ : (−∞, 1] −→ M (in other words, if (M, F )is forward complete), then we say that (M, F ) is backward complete. The backward
completeness is not necessarily equivalent to the forward completeness (in the noncompactcase). If (M, F ) is forward or backward complete, then the Hopf–Rinow theorem ensuresthat any pair of points is connected by a minimal geodesic (see [BCS, Theorem 6.6.1]).
7

2.3

Covariant derivative

We saw in (2.4) and (2.5) that the Finsler geodesic equation has a similar form to the
Riemannian one. This is, however, a special feature of the geodesic equation and we needto take care of some non-Riemannian quantities in the more general covariant derivative.
A fine property of the geodesic equation could be understood from Theorem 2.5. In order
to apply Theorem 2.5 to some quantity at v ∈ T M \ 0, we need a contraction with respect
to v. In the geodesic equation (2.4) we have the contractions twice thanks to η̇ j (t) andη̇ k (t), this procedure kills all the error (non-Riemannian) terms.
Notice that the geodesic equation for the Riemannian structure gη̇ (defined only alongη) coincides with the geodesic equation (2.4) with respect to F . This kind of property isextremely useful when we try to apply the techniques in Riemannian geometry to Finsler
geometry. This viewpoint leads us to modify the formal Christoﬀel symbol (2.5) intoi(v) −Γijk (v) := γjk

n
∑
g il

F
l,m=1

(Alkm Njm + Ajlm Nkm − Ajkm Nlm )(v),

where
1 ∂Gi
Nji (v) :=
(v),
j
2 ∂v

i

G (v) :=n∑v ∈ T M \ 0,

(2.6)

i
γjk
(v)v j v kj,k=1(the validity of the definition (2.6) can be seen in Proposition 2.6 below). We call Gi
and Nji the geodesic spray coeﬃcients and the nonlinear connection, respectively, and set
Gi (0) = Nji (0) := 0 by convention. Notice that (2.3) yieldsn∑Γijk (v)v j v k =

n
∑

i
γjk
(v)v j v k .

j,k=1

j,k=1

However, when we contract in v only once,n∑i{Γijk (v) − γjk(v)}v j = −

j=1

n
∑
g il (v)

F (v)
l,m=1

Alkm (v)Gm (v)does not necessarily vanish.
Define the covariant derivative of a vector field X by v ∈ Tx M with the reference
vector w ∈ Tx M \ 0 as
}
n {
n
i
∑
∑
∂
j ∂X
j k
w
i
v
Dv X(x) :=(x) +Γjk (w)v X (x)∈ Tx M.
(2.7)
j
∂x
∂xi x
i,j=1
k=1
With this definition, we can show the following important property (see [Sh2, §6.2] or[Oh8, Lemma 2.3]).
Proposition 2.6 Let V be a non-vanishing C ∞ -vector field on an open set U such that
all integral curves are geodesic. Then we have, for any C 1 -vector field W on U ,DVV W = DVgV W,

gVV
V = DW
DW
V.

Here gV is the Riemannian structure of U induced from V as (2.2) and DgV denotes its
corresponding covariant derivative.
8

In particular, integral curves of V are geodesic also with respect to gV . Note thatV
V . The geodesic
we have only one contraction (with respect to V ) in DVV W and DWη̇equation Dη̇ η̇ ≡ 0 is concerned with the special case of W = V , where we have one more
contraction.

2.4

Curvatures

In order to define the curvature, we again look at the behavior of geodesics. It can beshown that the variational vector field J of a geodesic variation σ : [0, 1] × (−ε, ε) −→ M
(i.e., σ(·, s) is geodesic for all s and J(t) := ∂σ/∂s(t, 0)) satisfies the Jacobi equationDη̇η̇ Dη̇η̇ J + Rη̇ (J) = 0(2.8)

and vice versa, where η(t) := σ(t, 0) and
n
∑

∂
,
i x
∂x
i,j=1
} ∑
n {
n
∑
∂Nji
∂Nji
∂Gi
i
k
k
Rj (v) :=(v) −(v)v −(v)G (v) −Nki (v)Njk (v).
k
k
∂xj
∂x
∂v
k=1
k=1

Rv (w) :=Rji (v)wj

We refer to [Sh2, §6.1] for details (where Gi is one-half of ours, while Nji is the same). It
is unnecessary to worry about the complicated formula of Rji (v), what we essentially need
will be only the Jacobi equation (2.8) and the characterization in Theorem 2.10 below.
Definition 2.7 (Flag curvature) For linearly independent vectors v, w ∈ Tx M , we define the flag curvature byK(v, w) :=gv (Rv (w), w).
2
F (v)gv (w, w) − gv (v, w)2

On Riemannian manifolds, the flag curvature K(v, w) coincides with the sectional
curvature of the 2-plane v ∧ w spanned by v and w. We remark that K(v, w) depends
not only on the flag v ∧ w, but also on the choice of the pole v in it. Thus, for example,
K(w, v) may be diﬀerent from K(v, w). We further define the Ricci curvature as follows.
Definition 2.8 (Ricci curvature) For a unit vector v ∈ Ux M , we define the Ricci
curvature as the trace of the flag curvature with respect to gv , namely
Ric(v) :=n−1∑

K(v, ei ),i=12where {ei }n−1i=1 ∪ {v} is orthonormal with respect to gv . We also define Ric(cv) := c Ric(v)for c ≥ 0.

As usual, given K ∈ R, the bound Ric ≥ K will mean that Ric(v) ≥ KF 2 (v) holds←−
for all v ∈ T M . We remark that this curvature bound is common to F and F since one←−
can see Ric(v) = Ric(−v) from Theorem 2.10 (recall Definition 2.4 for the definitions of←−←−
F and Ric).
9

Remark 2.9 The appropriate notion of scalar curvature is still missing in the Finsler
setting. This is one of the main obstructions to develop the theory of Finsler–Ricci flow.
Comparing Proposition 2.6 and (2.8), one finds the following quite important anduseful property.
Theorem 2.10 (A Riemannian characterization of Finsler curvatures) Given aunit vector v ∈ Ux M , take a C ∞ -vector field V on a neighborhood U of x such that
V (x) = x and all integral curves of V are geodesic. Then, for any w ∈ Tx M linearly independent from v, the flag curvature K(v, w) coincides with the sectional curvature of the
2-plane v ∧ w with respect to the Riemannian metric gV on U . Similarly, Ric(v) coincides
with the Ricci curvature of v with respect to gV .
We in particular find that the sectional curvature of v ∧ w with respect to gV does notdepend on the choice of V , where the assumption on V (all integral curves are geodesic)
is playing the essential role. The proof of Theorem 2.10 can be found in [Sh2, §6], which
inspired the structure of this section. The characterization in the manner of Theorem 2.10
goes back to at least [Au].
Via Theorem 2.10, one can reduce some results on Finsler manifolds to the Riemanniancase. The most fundamental ones are the following (obtained in [Au]), we will see somemore (concerning the weighted Ricci curvature) in the next subsection.
Theorem 2.11 (Cartan–Hadamard theorem) Let (M, F ) be a simply-connected andforward complete Finsler manifold with nonpositive flag curvature K ≤ 0. Then the
exponential map expx : Tx M −→ M is a C 1 -diﬀeomorphism for every x ∈ M .
Theorem 2.12 (Bonnet–Myers theorem) If (M, F ) is forward complete and satisfies
Ric ≥ K for some K > 0, then we have√n−1diam(M ) := sup d(x, y) ≤ π
.
K
x,y∈M
In particular, M is compact.
Outline of proof. Let us outline the proof of Theorem 2.12 (Theorem 2.11 is shown in asimilar manner). Fix a point x ∈ M and consider unit speed geodesics η : [0, lη ) −→ Memanating from x. Here lη ∈ (0, ∞) is taken as the supremum of t > 0 such thatd(x, η(t)) = t. Define the vector field V by V (η(t)) := η̇(t) for each η and t ∈ (0, lη ).
Then V is a C ∞ -vector field on an open set U ⊂ M \{x} and all integral curves are geodesic.
Thanks to Theorem 2.10, the Riemannian metric gV has the Ricci curvature √≥ K. Therefore the Riemannian Bonnet–Myers theorem applies and we have lη ≤ π (n − 1)/K.
This shows the claim.
2

2.5

Weighted Ricci curvature

In order to discuss further applications of the Ricci curvature, it is natural to employ asuitable measure on M . At this point, our Finsler setting has another diﬃculty that the10choice of a canonical measure is not unique. There are several known ways to generalize
the Riemannian volume measure, canonical in their own rights, leading to the diﬀerent
measures. Among others the most fundamental examples of constructive measures are
the Busemann–Hausdorﬀ measure and the Holmes–Thompson measure (see [AT]).
Our standpoint is that, instead of starting from a constructive measure, we consider
an arbitrary measure (this turns out natural, see Remark 2.15). Then, inspired by Theorem 2.10 as well as the theory of weighted Ricci curvature (also called the Bakry–Émery–
Ricci curvature) of Riemannian manifolds, one can define the weighted Ricci curvature asin [Oh3]. From here on, we fix a positive C ∞ -measure m on M (meaning that it is written
down as m = Φ dx1 · · · dxn in each local coordinates with some positive C ∞ -function Φ).
Definition 2.13 (Weighted Ricci curvature) Given a unit vector v ∈ Ux M , let V
be a non-vanishing C ∞ -vector field on a neighborhood U of x such that V (x) = v andall integral curves of V are geodesic. We decompose m as m = e−Ψ volgV on U , whereΨ ∈ C ∞ (U ) and volgV is the volume measure of gV . Denote by η : (−ε, ε) −→ M thegeodesic such that η̇(0) = v. Then, for N ∈ (−∞, 0) ∪ (n, ∞), define
RicN (v) := Ric(v) + (Ψ ◦ η)′′ (0) −(Ψ ◦ η)′ (0)2.
N −n

We also define as the limits:
Ric∞ (v) := Ric(v) + (Ψ ◦ η)′′ (0),Ricn (v) := lim RicN (v).
N ↓nFinally, for c ≥ 0, we set RicN (cv) := c2 RicN (v).
We will denote by RicN ≥ K, K ∈ R, the condition RicN (v) ≥ KF 2 (v) for all v ∈ T M .
Some remarks on Definition 2.13 are in order.
Remark 2.14 (a) In local coordinates, volgV along η is written as√( )det(gij η̇) dx1 · · · dxn .
Hence we find(√Ψ(η) = log)
det(gij (η̇)),Φ(η)

where m = Φ dx1 · · · dxn .

This expression does not require the vector field V (though we introduced it for the sakeof lucidity), thereby Ψ ◦ η does not depend on the choice of V .
(b) On a Riemnnian manifold (M, g), the metric gV always coincides with the original
metric g and hence Ψ is regarded as a function on M such that m = e−Ψ volg . If we choosem = volg , then RicN = Ric regardless of the choice of N .
(c) Multiplying the measure m with a positive constant does not change the weighted
Ricci curvature. Therefore we can freely normalize m as m(M ) = 1 when m(M ) < ∞.
(d) Notice that RicN is non-decreasing in N in the ranges [n, ∞] and (−∞, 0), and we
have
Ricn ≤ RicN ≤ Ric∞ ≤ RicN ′for N ∈ (n, ∞), N ′ < 0.
11

Therefore, RicN ′ ≥ K with N ′ < 0 is a weaker condition than RicN ≥ K for N ∈ (n, ∞).
In the Riemannian case, the study of Ric∞ goes back to Lichnerowicz [Li], he showed a
Cheeger–Gromoll type splitting theorem (see [FLZ, WW, Oh8] for some generalizations).
The range N ∈ (n, ∞) has been well investigated by Bakry [Bak, §6], Qian [Qi] andmany others. The study of the range N ∈ (−∞, 0) is more recent. We refer to [Mi2]for isoperimetric inequalities, [Oh9, Oh10] for the curvature-dimension condition (withN ≤ 0), and [Wy] for splitting theorems (with N ∈ (−∞, 1]). Some historical accounts
on related works concerning N < 0 in convex geometry and partial diﬀerential equations
can be found in [Mi2, Mi3].
Related to (b) above, a reason why we consider it natural to begin with an arbitrary
measure is explained as follows.
Remark 2.15 (S-curvature) For Finsler manifolds of Berwald type (i.e., Γkij is constanton each Tx M \ 0), the Busemann–Hausdorﬀ measure satisfies (Ψ ◦ η)′ ≡ 0 (in other words,
the S-curvature vanishes, see [Sh2, §7.3]). In general, however, there may not exist
any measure whose S-curvature vanishes (see [Oh5] for such an example among Randers
spaces). We regard this fact as the non-existence of a canonical measure, thus we beganwith an arbitrary measure.
The weighted version of the Bonnet–Myers theorem (Theorem 2.12) can be shown inthe same manner as the unweighted case.
Theorem 2.16 (Bonnet–Myers theorem) If (M, F, m) is forward complete and satisfies RicN ≥ K for some K > 0 and N ∈ [n, ∞), then we have√N −1diam(M ) ≤ π.
K
In particular, M is compact.
From this estimate N can be regarded as (an upper bound of) the dimension (though
this interpretation prevents us from considering N < 0). We can further control thevolume growth for m. For x ∈ M and r > 0, we define
B + (x, r) := {y ∈ M | d(x, y) < r}.
We also introduce the function sκ defined by√
1
if κ > 0, √κ sin( κθ)θ√if κ = 0,sκ (θ) := √1sinh( −κθ) if κ < 0,−κ√
where θ ≥ 0 if κ ≤ 0 and θ ∈ [0, π/ κ] if κ > 0.
Theorem 2.17 (Bishop–Gromov volume comparison) Suppose that (M, F, m) is
forward complete and satisfies RicN ≥ K for some K ∈ R and N ∈ [n, ∞). Then
we have∫R
sK/(N −1) (t)N −1 dtm(B + (x, R))0∫≤rm(B + (x, r))s(t)N −1 dt
0 K/(N −1)
12√for any x ∈ M and 0 < r < R, where R ≤ π (N − 1)/K if K > 0.
√
Note that the condition R ≤ π (N − 1)/K for K > 0 does not lose any generality because of Theorem 2.16. Similarly to Theorems 2.11, 2.12, 2.16, the proof of Theorem 2.17
can be reduced to the (weighted) Riemannian situation by employing the Riemannian
metric induced from the unit speed geodesics emanating from x.
Remark 2.18 (Curvature-dimension condition) The original motivation of introducing RicN in [Oh3] was to study Lott, Sturm and Villani’s curvature-dimension condition CD(K, N ) on Finsler manifolds (see also the surveys [Oh4, Oh6]). The celebrated
condition CD(K, N ) was introduced as a synthetic geometric notion playing a role ofa lower Ricci curvature bound for metric measure spaces ([St1, St2, LV]). On a complete Riemannian manifold with an arbitrary measure (M, g, m), the condition CD(K, N )is indeed equivalent to RicN ≥ K (see [vRS, St1, St2, LV], where N ∈ [n, ∞]). In[Oh3] this equivalence was generalized to Finsler manifolds (M, F, m), with applications
including the Bishop–Gromov volume comparison and some functional inequalities (seealso Section 5 for the latter). As we have already mentioned in Remark 2.14(d), thecurvature-dimension condition can be extended to N ∈ (−∞, 0] and is again equivalentto RicN ≥ K ([Oh9, Oh10]).
The theory of metric measure spaces satisfying CD(K, N ) is making a deep and breathtaking progress in this decade. We refer to Villani’s book [Vi] as a fundamental reference, and Cavalletti–Mondino’s Lévy–Gromov type isoperimetric inequalities ([CM1], seealso Section 6) as one of the most notable achievements after [Vi]. Since the condition
CD(K, N ) could not rule out Finsler manifolds, a reinforced version of CD(K, N ) coupledwith the linearity of the heat semigroup was introduced and is called the Riemanniancurvature-dimension condition RCD(K, N ) ([AGS, EKS]). In this Riemannian framework, many finer results such as Gigli’s splitting theorem [Gi1] and second order calculus[Gi2] as well as Mondino–Naber’s rectifiability [MN] are known.

3

Nonlinear Laplacian and heat semigroup

At this section we begin geometric analysis on Finsler manifolds. The main object of thesection is the Bochner–Weitzenböck formula (Theorem 3.3), which is the indispensable
ingredient of our nonlinear analogue of the Γ-calculus.

3.1

Gradient vectors, Hessian and Laplacian

Let us denote by L∗ : T ∗ M −→ T M the Legendre transform associated with F . That is
to say, L∗ is sending α ∈ Tx∗ M to the unique element v ∈ Tx M such that F (v) = F ∗ (α)
and α(v) = F ∗ (α)2 . In coordinates we can write downL∗ (α) =

n
∑
i,j=1

gij∗ (α)αjn∑∂1 ∂[(F ∗ )2 ]
∂
=
(α)
∂xi x
2 ∂αi
∂xi x
i=1

for α ∈ Tx∗ M \ 0 (the latter expression makes sense also at 0 as L∗ (0) = 0). Note that()
gij∗ (α) = g ij L∗ (α)

13for α ∈ Tx∗ M \ 0(recall that (g ij (v)) denotes the inverse matrix of (gij (v))). The map L∗ |Tx∗ M is being alinear operator only when F |Tx M comes from an inner product.
For a diﬀerentiable function u : M −→ R, the gradient vector at x is defined as the
Legendre transform of the derivative of u: ∇u(x) := L∗ (Du(x)) ∈ Tx M . If Du(x) ̸= 0,
then we can write down in coordinates as∇u(x) =

n
∑

(
) ∂u
∂
(x) i .
gij∗ Du(x)
j
∂x
∂x x
i,j=1

We need to be careful when Du(x) = 0, because gij∗ (Du(x)) is not defined as well as the
Legendre transform L∗ is only continuous at the zero section. Thus we set for later use
Mu := {x ∈ M | Du(x) ̸= 0}.
For a twice diﬀerentiable function u : M −→ R and x ∈ Mu , we define a kind of Hessian∇2 u(x) : Tx M −→ Tx M by using the covariant derivative (2.7) as∇2 u(v) := Dv∇u (∇u)(x) ∈ Tx M,

v ∈ Tx M.

The linear operator ∇2 u(x) is symmetric in the sense that()()
g∇u ∇2 u(v), w = g∇u v, ∇2 u(w)for all v, w ∈ Tx M with x ∈ Mu (see, for example, [OS3, Lemma 2.3]).
Define the divergence of a diﬀerentiable vector field V on M with respect to themeasure m by
)
n (
n
∑
∑
∂V i
i ∂Φ
i ∂
divm V :=
+
V
,
V
=
V
,
i
i
i
∂x
∂x
∂x
i=1
i=1
where we decomposed m as dm = eΦ dx1 dx2 · · · dxn . One can rewrite this in the weakform as∫∫ϕ divm V dm = −Dϕ(V ) dm
for all ϕ ∈ Cc∞ (M ),MMthat makes sense for measurable vector fields V with F (V ) ∈ L1loc (M ). Then we define1
the distributional Laplacian of u ∈ Hloc
(M ) by ∆u := divm (∇u) in the weak sense that∫∫ϕ∆u dm := −Dϕ(∇u) dm
for all ϕ ∈ Cc∞ (M ).
M

M

1
Notice that the space Hloc
(M ) is defined solely in terms of the diﬀerentiable structureof M . Since taking the gradient vector (more precisely, the Legendre transform L∗ ) is anonlinear operation, our Laplacian ∆ is nonlinear unless F is Riemannian.

Remark 3.1 (a) In the Riemannian case, the Laplacian ∆m associated with a measurem = eΦ volg can be written as∆m u = ∆g u + g(∇u, ∇Φ),14

where ∆g is the usual Laplacian with respect to g. Then we call ∆m the Witten Laplacianor the weighted Laplacian.
(b) It is also possible to define the Laplacian associated only with F , and regard our
Laplacian ∆ as the weighted one with respect to m. Such a definition can be found in[Lee] for instance (see also [Oh7]), however, it is more involved than our simple definition∆u = divm (∇u) that is also natural from the analytic viewpoint.

3.2

Bochner–Weitzenböck formula

Concerning the relation between the Laplacian and the Ricci curvature, it is not diﬃcult
to show the Laplacian comparison (by essentially reducing to the Riemannian situation
like Theorem 2.17, see [OS1]). This is regarded as an analytic counterpart of the directedversion of the Bishop–Gromov volume comparison (known as the measure contraction
property, see [Oh1, St2]).
Theorem 3.2 (Laplacian comparison) Let (M, F ) be forward complete, and assume
that RicN ≥ 0 for some N ∈ [n, ∞). Then, for any z ∈ M , the function u(x) := d(z, x)satisfies
N −1∆u(x) ≤d(z, x)point-wise on M \ ({z} ∪ Cutz ), and in the distributional sense on M \ {z}.
Denoted by Cutz is the cut locus of z, which is the set of cut points x = expz (v) suchthat η(t) = expz (tv) is minimal on [0, 1] but not minimal on [0, 1 + ε] for any ε > 0.
A more sophisticated application of our geometric analysis is the Bochner–Weitzenböck
formula. This yields the Bochner inequality, the heart of the Γ-calculus. The Finsler
versions of them were established in [OS3] (for N ∈ [n, ∞]) and [Oh11] (for N < 0). In
1
order to state the formula, we need the following notations. Given f ∈ Hloc
(M ) and ameasurable vector field V such that V ̸= 0 almost everywhere on Mf , we can define thegradient vector field and the Laplacian on the weighted Riemannian manifold (M, gV , m)
by
 n
∑
∂f ∂


g ij (V ) j i on Mf ,V∂x ∂x
∇ f :=∆V f := divm (∇V f ),
i,j=1


0
on M \ Mf ,where the latter is in the sense of distributions. We have ∇∇u u = ∇u and ∆∇u u = ∆u
1
1
for u ∈ Hloc
(M ) ([OS1, Lemma 2.4]). We also observe that, given u, f1 , f2 ∈ Hloc
(M ),
Df2 (∇∇u f1 ) = g∇u (∇∇u f1 , ∇∇u f2 ) = Df1 (∇∇u f2 )(3.1)

(to be precise, in (3.1) we replace ∇u with a non-vanishing vector field in a measurable
way on M \ Mu ).
Theorem 3.3 (Bochner–Weitzenböck formula) Given u ∈ C ∞ (M ), we have][ 2
∇u F (∇u)− D[∆u](∇u) = Ric∞ (∇u) + ∥∇2 u∥2HS(∇u)
∆
2
15

(3.2)

[

as well as∇u∆]
F 2 (∇u)(∆u)2− D[∆u](∇u) ≥ RicN (∇u) +2Nfor N ∈ (−∞, 0) ∪ [n, ∞] point-wise on Mu , where ∥ · ∥HS(∇u) denotes the Hilbert–Schmidt
norm with respect to g∇u .
In particular, if RicN ≥ K, then we have
]
[ 2
(∆u)2
∇u F (∇u)− D[∆u](∇u) ≥ KF 2 (∇u) +
∆
2
N

(3.3)

on Mu , that we will call the Bochner inequality. One can further generalize the Bochner–
Weitzenböck formula to a more general class of Hamiltonian systems (by dropping thepositive 1-homogeneity of F , see [Lee, Oh7]).
Remark 3.4 (F versus g∇u ) We stress that (3.2) cannot be reduced to the Bochner–
Weitzenböck formula of the Riemannian metric g∇u . In fact, in contrast to the identity∆∇u u = ∆u, RicN (∇u) does not coincide with the weighted Ricci curvature Ric∇u
N (∇u)of the weighted Riemannian manifold (M, g∇u , m). It is compensated in (3.2) by the factthat ∇2 u does not necessarily coincide with the Hessian of u with respect to g∇u (unless
all integral curves of ∇u are geodesic).
Recall that, even when u ∈ C ∞ (M ), ∇u is only continuous outside Mu . Thus we haveto pass to the integrated form, and it is done as follows ([OS3, Theorem 3.6], [Oh11]).
Theorem 3.5 (Integrated form) Assume RicN ≥ K for some K ∈ R and N ∈2(−∞, 0) ∪ [n, ∞]. Given u ∈ H01 (M ) ∩ Hloc
(M ) ∩ C 1 (M ) such that ∆u ∈ H01 (M ), we
have
])
(
[ 2
∫
∇u F (∇u)
dm
−
Dϕ ∇
2
M
{
}
∫
(∆u)2
2
≥
ϕ D[∆u](∇u) + KF (∇u) +
dm
(3.4)
N
M
1
for all nonnegative functions ϕ ∈ Hloc
(M ) ∩ L∞ (M ).

See the next subsection for the definition of H01 (M ). We see in Theorem 3.7(ii) belowthat global solutions (ut )t≥0 to the heat equation always enjoy the condition ut ∈ H01 (M )∩2Hloc(M ) ∩ C 1 (M ), and also ∆ut ∈ H01 (M ) when SF < ∞.

3.3

Heat equation

In [OS1, OS3], we have investigated the nonlinear heat equation ∂t u = ∆u associated with1
our Laplacian ∆. To recall some results in [OS1], we define the energy of u ∈ Hloc
(M ) by∫∫1
1
2
E(u) :=F (∇u) dm =F ∗ (Du)2 dm.
2 M
2 M
16

We remark that E(u) < ∞ does not necessarily imply E(−u) < ∞. Define H01 (M ) as theclosure of Cc∞ (M ) with respect to the (absolutely homogeneous) norm∥u∥H 1 := ∥u∥L2 + {E(u) + E(−u)}1/2 .
Note that (H01 (M ), ∥ · ∥H 1 ) is a Banach space.
Definition 3.6 (Global solutions) We say that a function u on [0, T ] × M , T > 0, isa global solution to the heat equation ∂t u = ∆u if it satisfies the following:()()(1) u ∈ L2 [0, T ], H01 (M ) ∩ H 1 [0, T ], H −1 (M ) ;(2) We have∫∫ϕ · ∂t ut dm = −Dϕ(∇ut ) dm
MM
for all t ∈ [0, T ] and ϕ ∈ Cc∞ (M ), where we set ut := u(t, ·).
We refer to [Ev] for the notations as in (1). The test function ϕ in (2) can be taken fromH01 (M ). Global solutions are constructed as gradient curves of the energy functional E in
the Hilbert space L2 (M ). As for the regularity, since our Laplacian is locally uniformly
elliptic thanks to the strong convexity of F , the classical theory of partial diﬀerential
equations applies. We summarize the existence and regularity properties in the next
theorem (see [OS1, §§3, 4] for details, we remark that our C ∞ -smooth F and m clearlyenjoy the mild smoothness assumption in [OS1, (4.4)]).
Theorem 3.7 (i) For each initial datum u0 ∈ H01 (M ) and T > 0, there exists a uniqueglobal solution u = (ut )t∈[0,T ] to the heat equation, and the distributional Laplacian∆ut is absolutely continuous with respect to m for all t ∈ (0, T ).
2(ii) One can take the continuous version of a global solution u, and it enjoys the Hloc
1,αregularity in x as well as the C -regularity in both t and x. Moreover, ∂t u lies in1Hloc(M ) ∩ C(M ), and further in H01 (M ) if SF < ∞.
∪We remark that the usual elliptic regularity yields that u is C ∞ on t>0 ({t} × Mut ).
The proof of ∂t u ∈ H01 (M ) under SF < ∞ can be found in [OS1, Appendix A]. It was also
proved in [OS1] that the heat flow is regarded as the gradient flow of the relative entropy←−(see the proof of Proposition 5.3 below) in the L2 -Wasserstein space with respect to F
(this result is far beyond the scope of the present survey).

4

Linearized heat semigroups and gradient estimates

In the Bochner–Weitzenböck formula (Theorem 3.3) in the previous section, we used thelinearized Laplacian ∆∇u induced from the Riemannian structure g∇u . In the same spirit,
we can consider the linearized heat semigroup associated with a global solution to theheat equation. This technique turns out useful, and we obtain some gradient estimatesas our first applications of the nonlinear Γ-calculus. We will find that, from here on, ourarguments rely only on the Bochner inequality, the integration by parts, and the (nonlinear
and linearized) heat semigroups. Calculations in local coordinates do not appear.
17

4.1

Linearized heat semigroups and their adjoints

Let (ut )t≥0 be a global solution to the heat equation. We will fix a measurable oneparameter family of non-vanishing vector fields (Vt )t≥0 such that Vt = ∇ut on Mut foreach t ≥ 0.
∇u
Given f ∈ H01 (M ) and s ≥ 0, let (Ps,t(f ))t≥s be the weak solution to the linearizedheat equation:∇u∇u∇u∂t [Ps,t(f )] = ∆Vt [Ps,t(f )],
Ps,s(f ) = f.
(4.1)∇u
The existence and other properties of the linearized semigroup Ps,t
are summarized asfollows ([OS1, Oh11]).

Proposition 4.1 (Properties of linearized semigroups) Assume SF < ∞, and let(ut )t≥0 and (Vt )t≥0 be as above.
∇u(i) For each s ≥ 0, T > 0 and f ∈ H01 (M ), there exists a unique solution ft = Ps,t(f ),t ∈ [s, s + T ], to (4.1) in the weak sense that
∫ s+T ∫
∫ s+T ∫∂t ϕt · ft dm dt =Dϕt (∇Vt ft ) dm dt
s

M

s

M

for all ϕ ∈ Cc∞ ((s, s + T ) × M ).
2(ii) The solution (ft )t∈[s,s+T ] is Hölder continuous on (s, s + T ) × M as well as Hloc
and
1,α
1
C in x. Moreover, we have ∂t ft ∈ H0 (M ) for t ∈ (s, s + T ).

The uniqueness in (i) follows from ∂t (∥ft ∥2L2 ) = −4E Vt (ft ) ≤ 0, where E Vt is the∇uenergy form with respect to gVt . This also implies that Ps,tuniquely extends to the2linear contraction semigroup acting on L (M ). Notice also that
( ∪
)
∞
f ∈C
({t} × Mut ) .
s<t<s+T∇u
The operator Ps,tis linear but asymmetric with respect to the L2 -inner product. Let∇u∇uus denote by Pbs,t
the adjoint operator of Ps,t
. That is to say, given h ∈ H01 (M ) and∇ut > 0, we define (Pbs,t(h))s∈[0,t] as the solution to the equation∇u∇u∂s [Pbs,t(h)] = −∆Vs [Pbs,t
(h)],

∫

Note that∇uh · Ps,t(f ) dm =

M

∫

∇u
Pbt,t
(h) = h.

∇uPbs,t(h) · f dm
Mindeed holds since for r ∈ (0, t − s)
[∫
]
∇u
∇u
∂r
Pbs+r,t (h) · Ps,s+r (f ) dm
M
∫
∫
∇u
∇u
∇u
Vs+r b ∇u(f )] dm(h) · ∆Vs+r [Ps,s+r
Pbs+r,t=−∆ [Ps+r,t (h)] · Ps,s+r (f ) dm +
M

M

= 0.
18

(4.2)

(4.3)

One may rewrite (4.2) as∇u∇u∂σ [Pbt−σ,t(h)] = ∆Vt−σ [Pbt−σ,t(h)],σ ∈ [0, t],

to see that the adjoint heat semigroup solves the linearized heat equation backward intime. (This evolution is sometimes called the conjugate heat semigroup, especially in the
Ricci flow theory, see for instance [Ch+ , Chapter 5].) Therefore we find in the same way∇u∇u
as Ps,tthat Pbt−σ,textends to the linear contraction semigroup acting on L2 (M ).
∇u∇umay depend on the choice of an
and Pbs,tRemark 4.2 In general, the semigroups Ps,tauxiliary vector field (Vt )t≥0 . This issue, however, will not aﬀect our discussions.

4.2

Gradient estimates

We deduce from the Bochner inequality (3.3) with N = ∞ the L2 -gradient estimate forheat flow. The proof is a good example of a typical argument in the Γ-calculus.
Theorem 4.3 (L2 -gradient estimate) Assume Ric∞ ≥ K and SF < ∞. Then, given
any global solution (ut )t≥0 to the heat equation with u0 ∈ Cc∞ (M ), we have
(
)
( 2
)
∇u
F 2 ∇ut (x) ≤ e−2K(t−s) Ps,tF (∇us ) (x)for all 0 ≤ s < t < ∞ and x ∈ M .
Let us stress that we used the nonlinear semigroup (us → ut ) in the LHS, while in the∇u
RHS the linearized semigroup Ps,tshows up.
Outline of proof. We remark that the condition u0 ∈ Cc∞ (M ) ensures F 2 (∇u0 ) ∈ Cc1 (M ),and hence both sides in Theorem 4.3 are Hölder continuous.
For fixed t > 0 and an arbitrary nonnegative function h ∈ C(M ) of compact support,
we set∫F 2 (∇us )2Ks∇u
H(s) := e
Pbs,t
(h) ·
dm,
0 ≤ s ≤ t.
2
M∇u
Then we deduce from the definition of Pbs,tthat)
F 2 (∇us ) ( ∇us ( b∇u ))∇Ps,t (h) dm
H (s) = e
D
2
M
(
)
∫
∂ F 2 (∇us )∇u
2Ksb
Ps,t (h) ·+edm + 2KH(s).
∂s
2
M
′

∫

(

2Ks

By (3.1), the first term in the RHS coincides with
(
( 2
))
∫
( ∇u )∇us F (∇us )2Ksbe
D Ps,t (h) ∇
dm.
2
M

19

The other terms are calculated with the help of Theorem 2.5 as (see [OS3] for details)
{ (
)
}
∫
∂us
2Ks
∇u
2
b
e
Ps,t (h) D(∇us ) + KF (∇us ) dm
∂s
M
∫
{
}
2Ks
∇u
=e
Pbs,t
(h) D(∆us )(∇us ) + KF 2 (∇us ) dm.
M

Therefore we have H ′ (s) ≤ 0 by the Bochner inequality (3.4).
Thus we find H(t) ≤ H(s) for any 0 ≤ s ≤ t, which yields via (4.3)∫∫
F 2 (∇ut )F 2 (∇us )
2Kt
2Ks
∇u
e
h
dm ≤ e
Pbs,t
(h) ·
dm
2
2
M
M
)
( 2
∫
2Ks
∇u F (∇us )=eh · Ps,tdm.
2MSince this holds true for any nonnegative h, we obtain the claimed inequality for almost
all x ∈ M . This completes the proof since both sides are Hölder continuous.
2
In the linear setting, it is known that the Bochner inequality enjoys a self-improving
property ([BQ, Theorem 6], [BGL]). In our Finsler setting, however, the self-improvement
works only when F is reversible (as far as the author knows). Nonetheless, we can show
the improved inequality by the direct calculation ([Oh11]).
Proposition 4.4 (Improved Bochner inequality) Assume Ric∞ ≥ K for some K ∈2
R. Given u ∈ H01 (M ) ∩ Hloc
(M ) ∩ C 1 (M ) such that ∆u ∈ H01 (M ), we have
])
(
[ 2
∫
∇u F (∇u)
dm
−
Dϕ ∇
2
M
∫
{
( ∇u
)}
2
≥
ϕ D[∆u](∇u) + KF (∇u) + D[F (∇u)] ∇ [F (∇u)] dm
M
1
for all nonnegative functions ϕ ∈ Hloc
(M ) ∩ L∞ (M ).

By using this improved inequality instead of (3.4), we have the L1 -gradient estimate(which implies the L2 -gradient estimate via Jensen’s inequality, see [Oh11]).
Theorem 4.5 (L1 -gradient estimate) Assume Ric∞ ≥ K, SF < ∞ and the completeness of (M, F ). Then, given any global solution (ut )t≥0 to the heat equation withu0 ∈ Cc∞ (M ), we have
(
)
(
)
∇u
F (∇us ) (x)F ∇ut (x) ≤ e−K(t−s) Ps,tfor all 0 ≤ s < t < ∞ and x ∈ M .
Recall from Lemma 2.3 that SF < ∞ implies the finite reversibility ΛF < ∞. Hence←−F and F are comparable and the forward and backward completenesses are mutuallyequivalent. Thereby we called it the plain completeness in the theorem above.

20

4.3

Characterizations of lower Ricci curvature bounds

We saw in the previous subsection that the Bochner inequalities imply the gradient estimates. The converse also holds true and, furthermore, they characterize the lower Ricci
curvature bound Ric∞ ≥ K as follows ([Oh11]).
Theorem 4.6 (Characterizations of Ricci curvature bounds) Suppose that SF <∞ and (M, F ) is complete. Then, for each K ∈ R, the following are equivalent:(I) Ric∞ ≥ K.
(II) The Bochner inequality
∇u

∆

[

]
F 2 (∇u)− D[∆u](∇u) ≥ KF 2 (∇u)
2

holds on Mu for all u ∈ C ∞ (M ).
(III) The improved Bochner inequality
]
[ 2
(
)
∇u F (∇u)∆− D[∆u](∇u) − KF 2 (∇u) ≥ D[F (∇u)] ∇∇u [F (∇u)]
2
holds on Mu for all u ∈ C ∞ (M ).
(IV) The L2 -gradient estimate

( 2
)
∇u
F 2 (∇ut ) ≤ e−2K(t−s) Ps,tF (∇us ) ,0 ≤ s < t < ∞,holds for all global solutions (ut )t≥0 to the heat equation with u0 ∈ Cc∞ (M ).
(V) The L1 -gradient estimate()∇u
F (∇ut ) ≤ e−K(t−s) Ps,tF (∇us ) ,0 ≤ s < t < ∞,holds for all global solutions (ut )t≥0 to the heat equation with u0 ∈ Cc∞ (M ).
In the Riemannian setting, one can add to this list the contraction property of heatflow with respect to the L2 -Wasserstein distance W2 (see [vRS, Ku] for details):
W2 (ut m, ūt m) ≤ e−Kt W2 (u0 m, ū0 m)∫∫
for global solutions (ut )t≥0 , (ūt )t≥0 such that u0 , ū0 ≥ 0 as well as M u0 dm = M ū0 dm =1. In our Finsler situation, however, the contraction property is known to fail ([OS2]).

5Functional inequalities

In this section we discuss three functional inequalities obtained in [Oh12]. The proofs
are based on the Γ-calculus, we refer to the book [BGL] for the usual linear setting. In
the reversible case, there is another technique called the localization which gives the same
sharp estimates even in a more general framework of essentially non-branching metricmeasure spaces satisfying the curvature-dimension condition (see [CM2]). In the nonreversible case, however, the localization seems to give only non-sharp estimates, see §6.1
for a more detailed discussion.
21

5.1

Poincaré–Lichnerowicz inequality

We start with the Poincaré–Lichnerowicz (spectral gap) inequality under the curvaturebound RicN ≥ K > 0. The N ∈ [n, ∞] case was shown in [Oh3] as a consequence ofthe curvature-dimension condition CD(K, N ). In the Riemannian setting, the case ofN ∈ (−∞, 0) was shown independently in [KM] and [Oh9].
For simplicity we will assume that M is compact (this is automatically true whenN ∈ [n, ∞) and M is complete), and normalize m as m(M ) = 1. We will give the proofs
since they are not long.
Proposition 5.1 Assume that M is compact and satisfies RicN ≥ K for some K ∈ R
and N ∈ (−∞, 0) ∪ [n, ∞]. Then we have, for any f ∈ H 2 (M ) ∩ C 1 (M ) such that∆f ∈ H 1 (M ),}∫ {(∆f )2
2
D[∆f ](∇f ) + KF (∇f ) +dm ≤ 0.
N
M
In particular, if K > 0, then we have
∫
∫
N −1
2
F (∇f ) dm ≤(∆f )2 dm.
KN M
M

(5.1)

If N = ∞, then the coeﬃcient in the RHS of (5.1) is read as 1/K.
Proof. The first assertion is a direct consequence of Theorem 3.5 with ϕ ≡ 1. We further
observe, by the integration by parts,∫∫∥∆f ∥2L2
2
K
F (∇f ) dm +≤−D[∆f ](∇f ) dm = ∥∆f ∥2L2 .
N
M
M
Rearranging this inequality yields (5.1) when K > 0.

2

Now the Poincaré–Lichnerowicz inequality is obtained by a technique somewhat related to the proof of Theorem 4.3. We define the variance of f ∈ L2 (M ) (under m(M ) = 1)
as
(∫
)2
∫
2
f dm .
Varm (f ) :=f dm −
M

M

Theorem 5.2 (Poincaré–Lichnerowicz inequality) Let M be compact, m(M ) = 1,
and suppose RicN ≥ K > 0 for some N ∈ (−∞, 0) ∪ [n, ∞]. Then we have, for any
f ∈ H 1 (M ),∫N −1
F 2 (∇f ) dm.
Varm (f ) ≤KN M
Proof. Let (ut )t≥0 be the solution to the heat equation with u0 = f , and put Φ(t) :=∥ut ∥2L2 for t ≥ 0. Observe first that the ergodicity∫
ut →u0 dm in L2 (M )
(5.2)
M

22

holds since limt→∞ E(ut ) = 0. It follows from the heat equation that∫∫′ut ∆ut dm = −2
F 2 (∇ut ) dm = −4E(ut ),Φ (t) = 2M

as well asMΦ′ (t + δ) − Φ′ (t)E(ut+δ ) − E(ut )= −4 lim= 4∥∆ut ∥2L2
δ↓0
δ↓0
δ
δ

lim

for all t > 0. Then (5.1) implies
N −1
Φ′ (t + δ) − Φ′ (t)
lim
.
(5.3)
KN δ↓0δ∫
Notice also that the ergodicity (5.2) implies limt→∞ Φ(t) = ( M f dm)2 . Thus the diﬀerential inequality (5.3) yields
(
)
∫ ∞
N −1
2(N − 1)′′′
Varm (f ) = −Φ (t) dt ≤lim Φ (t) − Φ (0) =E(f ).
2KN t→∞
KN
0−2Φ′ (t) ≤2This completes the proof.

5.2

Logarithmic Sobolev inequality

We next study the logarithmic Sobolev inequality. From here on we consider only N ∈[n, ∞) with K > 0 (actually the inequality fails for N < 0, see Remark 5.6 below). See[Oh3] for the case of N = ∞. Recall from Theorem 2.16 that RicN ≥ K implies thecompactness of M , thus we normalize m as m(M ) = 1 without loss of generality. We firstconsider a suﬃcient condition for the logarithmic Sobolev inequality.
Proposition 5.3 Assume that M is compact, Ric∞ ≥ K > 0, and
])
}
[ 2
∫ { (
∫
F 2 (∇u)∇u F (∇u)
dm ≤ −CDu ∇+ uD[∆(log u)](∇[log u]) dm (5.4)
u
2u2
M
M
holds for some constant C > 0 and all functions u ∈ H 2 (M ) ∩ C 1 (M ) such that ∆u ∈H 1 (M ) and inf M u > 0. Then the logarithmic Sobolev inequality∫∫
F 2 (∇f )C
dm
f log f dm ≤2 {f >0}f{f >0}∫
holds for all nonnegative functions f ∈ H 1 (M ) with M f dm = 1.
Outline of proof. We shall assume inf M f > 0 and use a similar method to the proof of
Theorem 5.2 for the relative entropy∫
Ψ(t) = Entm (ut m) :=ut log ut dm,
M23where (ut )t≥0 is the solution to the heat equation with u0 = f . The inequality (5.4)
then yields the diﬀerential inequality −2Ψ′ (t) ≤ CΨ′′ (t). Together with limt→∞ Ψ(t) =limt→∞ Ψ′ (t) = 0, we obtain∫∫ ∞∫C ∞ ′′
C′
f log f dm = −Ψ (t) dt ≤Ψ (t) dt = − Ψ′ (0).
2 0
2
M
0
2

This is indeed the desired inequality.

See [Oh12] and [BGL] for the omitted calculations in the proofs of the proposition
above and the theorem below.
Theorem 5.4 (Logarithmic Sobolev inequality) Assume that RicN ≥ K > 0 forsome N ∈ [n, ∞) and m(M ) = 1. Then we have∫∫
N −1
F 2 (∇f )f log f dm ≤dm2KN {f >0}
f{f >0}∫
for all nonnegative functions f ∈ H 1 (M ) with M f dm = 1.
Outline of proof. Fix h ∈ C ∞ (M ) and consider the function eah for a > 0 (chosen later).
For brevity let us introduce the notation common in the Γ-calculus:[ 2]∇h F (∇h)Γ2 (h) := ∆− D[∆h](∇h).
2
On the one hand, by the chain rule and a > 0, we calculate{}
Γ2 (eah ) = a2 e2ah Γ2 (h) + aD[F 2 (∇h)](∇h) + a2 F 4 (∇h)(notice that a > 0 yields ∇(eah ) = aeah ∇h). On the other hand, it follows from theintegration by parts that
∫
∫
{
}
ah
2
Γ2 (e ) dm = ae2ah (∆h)2 − 2aD[F 2 (∇h)](∇h) − 3a2 F 4 (∇h) dm.
M

MComparing these yields
∫
∫
{
}
2ah
2
e (∆h) dm =e2ah Γ2 (h) + 3aD[F 2 (∇h)](∇h) + 4a2 F 4 (∇h) dm.
M

(5.5)

M

We apply the Bochner inequality (3.3) to eah , now written as Γ2 (eah ) ≥ KF 2 (∇eah ) +(∆eah )2 /N , and seeΓ2 (h) + aD[F 2 (∇h)](∇h) + a2 F 4 (∇h)}1{(∆h)2 + 2aF 2 (∇h)∆h + a2 F 4 (∇h)
≥ KF 2 (∇h) +
N

24

(5.6)

in the weak sense. Integrating this inequality multiplied by the test function eh , using(5.5) with a = 1/2 and rearranging, we obtain
(
)∫
∫
1
h
1−
e Γ2 (h) dm ≥ K
eh F 2 (∇h) dm
NMM∫
3 − 2(N + 2)a+eh D[F 2 (∇h)](∇h) dm
2NM∫(a − 1)2 − N a2
eh F 4 (∇h) dm.
(5.7)
+
N
M
We finally choose a = 3/{2(N + 2)} > 0 and conclude that
∫
∫
N −1
h
e Γ2 (h) dm ≥ K
eh F 2 (∇h) dm.
N
M
M
This is the desired inequality (5.4) for u = eh . We complete the proof by an approximationargument.
2
By the standard implication going back to [OV], we have the Talagrand inequality asa corollary.
Corollary 5.5 (Talagrand inequality) Assume that RicN ≥ K > 0 for N ∈ [n, ∞)and m(M ) = 1. Then we have, for all µ ∈ P(M ),W22 (µ, m) ≤2(N − 1)
Entm (µ),
KN

where W2 is the L2 -Wasserstein distance.
Remark 5.6 Diﬀerent from the Poincaré–Lichnerowicz inequality in the previous section,
N cannot be negative in the logarithmic Sobolev and Talagrand inequalities. This isbecause the Talagrand inequality implies the normal concentration of m ([Led]), while
the model space in [Mi2, Oh9] enjoys only the exponential concentration. See [Ma] forfurther details.

5.3

Sobolev inequalities

Our next object is the Sobolev inequality. We will first obtain a non-sharp inequality
followed by some qualitative consequences. Then, with the help of those properties, we
proceed to the sharp estimate. The resulting inequality (Theorem 5.11) is the same as
the Riemannian case, however, we will need an additional care on the admissible range ofthe exponent p when F is non-reversible.
Proposition 5.7 (Logarithmic entropy-energy inequality) Assume RicN ≥ K > 0
for N ∈ [n, ∞) and m(M ) = 1. Then we have
(
)
∫
4
N
2
2
log 1 +
F (∇f ) dm
Entm (f m) ≤2
KN M∫
for all f ∈ H 1 (M ) with M f 2 dm = 1.
25

Outline of proof. It suﬃces to consider the case where c ≤ f ≤ C for some 0 < c <C < ∞. Let (ut )t≥0 be the solution to the heat equation with u0 = f 2 , and put Ψ(t) :=Entm (ut m) similarly to Proposition 5.3. By the Bochner inequality (3.3) and the Cauchy–
Schwarz inequality we findΨ′′ (t) ≥ −2KΨ′ (t) +2 ′ 2
Ψ (t) .
N

This implies that the function(−2Kt

t 7−→ e

1
K
− ′
N
Ψ (t)

)

is non-decreasing in t > 0, and hence
)
}−1
{
(
KN
2Kt
−1
.
−Ψ (t) ≤ KN e
1− ′
Ψ (0)
′

Integrating this inequality gives()′
N−2Kt Ψ (0)Ψ(0) − Ψ(t) ≤log 1 − (1 − e
)
.
2
KN
Since

∫

F 2 (∇(f 2 ))
Ψ (0) = −dm = −4
f2
M
letting t → ∞ completes the proof.
′∫
F 2 (∇f ) dm,
M

2

The above inequality yields the Nash inequality and then a non-sharp Sobolev inequality.
Lemma 5.8 (Nash inequality) Assume that RicN ≥ K > 0 for N ∈ [n, ∞) andm(M ) = 1. Then we have, for all f ∈ H 1 (M ),
)N/2
(
4
N +2
2
∥f ∥L2 ≤ ∥f ∥L2 +E(f )∥f ∥2L1 .
KN∫
Outline of proof. Normalize f so as to satisfy M f 2 dm = 1, and put ψ(θ) := log(∥f ∥L1/θ )for θ ∈ (0, 1]. We see by the Hölder inequality that ψ is a convex function. Therefore
( )
( )
( )
1
1 ′ 1
1 ′ 1
1
ψ(1) ≥ ψ+ ψ= ψ= − Entm (f 2 m).
2
2
2
2
2
2
Combining this with Proposition 5.7 gives the claim.

2

Proposition 5.9 (Non-sharp Sobolev inequality) Assume that RicN ≥ K > 0 forN ∈ [n, ∞) ∩ (2, ∞) and m(M ) = 1. Then we have∥f ∥2Lp ≤ C1 ∥f ∥2L2 + C2 E(f )for all f ∈ H 1 (M ), where p = 2N/(N − 2), C1 = C1 (N ) > 1 and C2 = C2 (K, N ) > 0.
26

Outline of proof. We can assume c ≤ f ≤ C for some 0 < c < C < ∞. Slice f intok
if f (x) > 2k+1 , 2
f − 2k if 2k < f (x) ≤ 2k+1 ,fk (x) :=
0
if f (x) ≤ 2k ,for k ∈ Z. We show the claim by applying the Nash inequality to each fk .

2

The above Sobolev inequality has some applications those will be used to show the
sharp inequality. Actually one can reduce such qualitative arguments to the Riemanniancase by observing the following.
Corollary 5.10 Assume that RicN ≥ K > 0 for N ∈ [n, ∞) ∩ (2, ∞) and m(M ) = 1.
Then there exists a C ∞ -Riemannian metric g for which∥f ∥2Lp ≤ C1 ∥f ∥2L2 + C2 SF E g (f )holds for all f ∈ H 1 (M ), where E g is the energy form of (M, g, m) and p = 2N/(N − 2),
C1 > 1 and C2 > 0 are as in Proposition 5.9.
We are now ready to show the sharp Sobolev inequality.
Theorem 5.11 (Sobolev inequality) Assume that RicN ≥ K > 0 for N ∈ [n, ∞) andm(M ) = 1. Then we have∫∥f ∥2Lp − ∥f ∥2L2
N −1
F 2 (∇f ) dm
≤
p−2
KN M
for all 1 ≤ p ≤ 2(N + 1)/N and f ∈ H 1 (M ).
The case of p = 2 is understood as the limit, giving the logarithmic Sobolev inequality (Theorem 5.4). The p = 1 case amounts to the Poincaré–Lichnerowicz inequality(Theorem 5.2).
Outline of proof. Take the smallest possible constant C > 0 satisfying∥f ∥2Lp − ∥f ∥2L2
≤ 2CE(f )
p−2

(5.8)

for all nonnegative functions f ∈ H 1 (M ). In order to show C ≤ (N − 1)/KN , let us
suppose that there is an extremal (nonconstant) function f ≥ 0 enjoying equality in (5.8)
as well as 0 < c ≤ f ≤ C < ∞. We normalize f as ∥f ∥Lp = 1. The equality in (5.8)
impliesf p−1 − f = −C(p − 2)∆f,which improves the regularity of f . Put u := log f .

27

On the one hand, for b ≥ 0, it follows from (5.5) (with a = b/2) and the integration
by parts that∫∫buCe Γ2 (u) dm =ebu F 2 (∇u) dm
M
M
(
)∫
b
+C p−1−
ebu D[F 2 (∇u)](∇u) dm
2∫M+ C(p − 2)(b − 1)
ebu F 4 (∇u) dm.
(5.9)
M

We remark that b ≥ 0 was required to apply (5.5). On the other hand, we deduce from
(5.6) that
)∫
(
∫
1
bu
e Γ2 (u) dm ≥ K
ebu F 2 (∇u) dm
1−
N
M
(M
)∫
3b − 4a+−aebu D[F 2 (∇u)](∇u) dm
2N
(
)M
∫
2
(a − b)2−aebu F 4 (∇u) dm,(5.10)+
N
M
which can be seen as a variant of (5.7). Comparing (5.9) and (5.10), we would like tochoose a and b enjoying
p−1−

3b − 2(N + 2)ab=,22(N − 1)(p − 2)(b − 1) =(a − b)2 − N a2
N −1

as well as a ≥ 0, b ≥ 0. This is possible for p ∈ [1, 2(N + 1)/N ], and then we concludeC ≤ (N − 1)/KN as desired.
There remains the delicate issue that such an extremal function f does not necessarily
exist. Hence one needs an extra discussion on the approximation. This technical argument
can be reduced to the Riemannian case thanks to the non-sharp Sobolev inequality in
Corollary 5.10. See [BGL, Theorem 6.8.3] for details.
2
Remark 5.12 The admissible range of the exponent p is [1, 2N/(N − 2)] in the Riemannian and the reversible Finsler cases. In the above non-reversible situation, the additional
constraints a ≥ 0 and b ≥ 0 make the range narrower. One can slightly extend the range[1, 2(N + 1)/N ] in Theorem 5.11 by a careful calculation into√[]
7N 2 + 2N + (N + 2) N 2 + 8N
1,,
4N (N − 1)though it is still smaller than [1, 2N/(N − 2)].

6

Gaussian isoperimetric inequality

This final section is devoted to a geometric application of the Γ-calculus, the Gaussian
isoperimetric inequality obtained in [Oh11]. This is the infinite dimensional counterpart28

to the Lévy–Gromov isoperimetric inequality, and was first established in the Riemanniansetting by Bakry–Ledoux [BL]. We will give historical background in §6.1 and the outlineof the proof in §6.2.

6.1

Background for Lévy–Gromov isoperimetic inequality

Given a Finsler manifold (M, F, m) such that m(M ) = 1, we define the isoperimetricprofile I(M,F,m) : [0, 1] −→ [0, ∞] by
I(M,F,m) (θ) := inf{m+ (A) | A ⊂ M : Borel set with m(A) = θ},
wherem+ (A) := lim inf
ε↓0m(B + (A, ε)) − m(A),εB + (A, ε) := {y ∈ M | inf d(x, y) < ε}.
x∈A

In the Riemannian case, the classical theorem of Lévy and Gromov [Lé1, Lé2, Gr] asserts
that, for an n-dimensional Riemannian manifold (M, g) of Ric ≥ n−1 with the normalized
volume measure m := volg (M )−1 volg , the isoperimetric profile I(M,g,m) is bounded frombelow by the profile of the unit sphere Sn :I(M,g,mg ) (θ) ≥ I(Sn ,mSn ) (θ),(6.1)

where mSn is the normalized volume measure as well. The standard strategy of the proofof (6.1) is as follows:(1) We take an extremal region A ⊂ M achieving the minimal boundary measure m+ (A)with the prescribed volume θ.
(2) Then the deep theorem in geometric measure theory (à la Federer, Almgren et al)
guarantees a certain regularity of the boundary of A.
(3) Perturbing A gives a diﬀerential inequality (a Heintze–Karcher type inequality).
(4) Combining the above diﬀerential inequality with the analysis of the behavior as θ ↓ 0gives the isoperimetic inequality.
Along the same strategy one can study the weighted version ([Bay]) and, moreover,the combinations of upper diameter bounds and lower curvature bounds ([Mi1]). The
most general work of Milman [Mi1] gave the sharp estimate:I(M,g,m) (θ) ≥ IK,N,D (θ)(6.2)

for (M, g, m) with RicN ≥ K and diam M ≤ D, where IK,N,D is the explicit function.
Up to now, however, the regularity theory is known only for Riemannian manifolds. This
had been an obstacle for generalizations to Finsler manifolds as well as less smooth spaces
such as metric measure spaces.
In 2014, Klartag [Kl] gave a beautiful alternative proof of the Lévy–Gromov isoperimetric inequality, still on weighted Riemannian manifolds, but without the regularity29

theory. His breakthrough was done by generalizing the localization method in convex
geometry to Riemannian manifolds with the help of optimal transport theory. The localization method, going back to [PW, GM, LS, KLS], is a sophisticated tool reducing an
inequality to those on geodesics. Then the analysis becomes much simpler and clearer.
Inspired by [Kl], Cavalletti–Mondino generalized the localization method to essentiallynon-branching metric measure spaces satisfying the curvature-dimension condition, andshowed the isoperimetric inequality (6.2) in [CM1] and several functional inequalities in[CM2]. This class of spaces includes reversible Finsler manifolds. In [Oh10], we generalized
the argument in [CM1] to possibly non-reversible Finsler manifolds, however, then it
turned out that the localization method gives only a non-sharp estimate in the nonreversible case, precisely,
I(M,F,m) (θ) ≥ Λ−1(6.3)
F · IK,N,D (θ).
This is due to the fact that reverse curves of geodesics are not necessarily geodesic. It
seems plausible to expect that Λ−1
F in (6.3) would be removed, that is to say, non-reversible
Finsler manifolds enjoy the same isoperimetric inequality as reversible Finsler manifolds.
Towards this direction, we consider the special case where N = D = ∞ and K > 0.
This is the only case where we have another alternative proof of (6.1), based on the Γcalculus ([BL]). As we saw in the previous section, the Γ-calculus is not sensitive to the
non-reversibility (only the exception was the range of p in Theorem 5.11), and we actuallyobtain the sharp estimate.

6.2

Isoperimetric inequality

In order to state the key estimate which is a kind of gradient estimate, we define∫ c12
φ(c) := √e−b /2 db for c ∈ R,
N (θ) := φ′ ◦ φ−1 (θ) for θ ∈ (0, 1).
2π −∞Set also N (0) = N (1) := 0.
Theorem 6.1 Assume Ric∞ ≥ K for some K ∈ R and SF < ∞. Then we have, given a
global solution (ut )t≥0 to the heat equation with u0 ∈ Cc∞ (M ) and 0 ≤ u0 ≤ 1,
(√
)
√
∇u
2
2
2
2
N (ut ) + αF (∇ut ) ≤ P0,tN (u0 ) + cα (t)F (∇u0 )(6.4)
for all α ≥ 0 and t > 0, wherecα (t) :=1 − e−2Kt+ αe−2Kt > 0Kand cα (t) := 2t + α when K = 0.
For simplicity, we suppressed the dependence of cα on K.
Outline of proof. By the construction of global solutions as gradient curves, we find that0 ≤ u0 ≤ 1 implies 0 ≤ ut ≤ 1 for all t > 0, and hence N (ut ) makes sense. Fix t > 0 andput√ζs := N 2 (us ) + cα (t − s)F 2 (∇us ),0 ≤ s ≤ t.
30

∇u
∇u
Then (6.4) is written as ζt ≤ P0,t(ζ0 ), thereby we are done when we show ∂s [Ps,t(ζs )] ≤ 0.
We can see by a simple calculation that∇u∇u∂s [Ps,t(ζs )] = Ps,t(∂s ζs − ∆∇us ζs ).

Hence it is suﬃcient to prove ∆∇us ζs − ∂s ζs ≥ 0 for 0 < s < t.
By using c′α (t) = 2(1 − Kcα (t)), N ′′ = −1/N and the improved Bochner inequality(Proposition 4.4), we obtain∆∇us ζs − ∂s ζs ≥cα (t − s)N ′ (us )2 4
F (∇us )ζs3( ∇us 2)cα (t − s)N (us )N ′ (us )
Du
−
∇
[F
(∇u
)]
s
s
ζs3
](
[ 2
])
[ 2
cα (t − s) N 2 (us )F (∇us )∇us F (∇us )
D
∇
.
+
ζs3
F 2 (∇us )2
2

Combining this with the Cauchy–Schwarz inequality
√
(
)
(
)
Dus ∇∇us [F 2 (∇us )] ≤ F (∇us ) D[F 2 (∇us )] ∇∇us [F 2 (∇us )]for g∇us , we conclude that∆∇us ζs − ∂s ζs
)
(
√
(
) 2
cα (t − s)N (us )
′
2
2
∇u
2
s
≥
D[F (∇us )] ∇ [F (∇us )]|N (us )|F (∇us ) −ζs3
2F (∇us )≥ 0.
2This completes the proof.

If K > 0, then Ric∞ ≥ K implies m(M ) < ∞ and hence we can normalize m (see[St1]). Choosing α = K −1 in (6.4) and letting t → ∞ yields the following.
Corollary 6.2 Assume that (M, F ) is complete and satisfies Ric∞ ≥ K > 0, SF < ∞and m(M ) = 1. Then we have, for any u ∈ Cc∞ (M ) with 0 ≤ u ≤ 1,
(∫
) ∫
√
√
u dm ≤KN
KN 2 (u) + F 2 (∇u) dm.
(6.5)
M

M

Now we are ready to show the isoperimetric inequality.
Theorem 6.3 (Gaussian isoperimetric inequality) Let (M, F, m) be complete andsatisfy Ric∞ ≥ K > 0, m(M ) = 1 and SF < ∞. Then we haveI(M,F,m) (θ) ≥ IK (θ)
for all θ ∈ [0, 1], where
IK (θ) :=√K −Kc2 (θ)/2
e
2π

(6.6)

∫ c(θ) √with θ =−∞31

K −Ka2 /2eda.
2πOutline of proof. Let θ ∈ (0, 1). Fix a closed set A ⊂ M with m(A) = θ and consideruε (x) := max{1 − ε−1 d(x, A), 0},

ε > 0.

Notice that F (∇uε ) = ε−1 on B − (A, ε) \ A, where{}B − (A, ε) := x ∈ M inf d(x, y) < ε .
y∈AApplying (6.5) to (smooth approximations of) uε and letting ε ↓ 0 implies, with the helpof N (0) = N (1) = 0,√m(B − (A, ε)) − m(A)KN (θ) ≤ lim inf
.
ε↓0
ε
←
−
This is the desired isoperimetric inequality for the reverse Finsler structure F (recall←−
Definition 2.4). Because the curvature bound Ric∞ ≥ K is common to F and F , we also
obtain (6.6).
2
The inequality (6.6) has the same form as the Riemannian case in [BL], thus it is sharp
and a √
model space is the real line R equipped with the normal (Gaussian) distribution2dm = K/2π e−Kx /2 dx. See [BL] for the original work on general linear diﬀusion semigroups (influenced by Bobkov’s works [Bob1, Bob2]), [Bor, SC] for the classical Euclidean
or Hilbert cases, and [AM] for the recent result on RCD(K, ∞)-spaces.

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