数理解析研究所講究録
第 1837 巻 2013 年 44-51

44

Curvature-dimension condition and heat flow
on metric measure spaces
Shin-ichi Ohta*
The curvature-dimension condition $CD(K, N)$ , introduced by Sturm and Lott-Villani,
is a generalized notion of the combination of a ‘lower Ricci curvature bound’ $(Ric\geq K)$
and an ‘upper dimension bound’ $(\dim\leq N)$ . Metric measure spaces satisfying $CD$ enjoy
many nice properties and well investigated from analytic and geometric points of view. In
this note we give a short review on $CD$ and heat flow on metric measure spaces satisfying$CD$ .
We refer to surveys [Lo], [Oh3] and the book [Vi2] for more about $CD$ , while thisnote is also concemed with more recent development.

1

Prehistory

We begin with some precursors of the curvature-dimension condition.

1.1

Wasserstein spaces

We need to review basic notions in optimal transport theory, for which we refer to [Vil]theand [Vi2]. Let ( $X$ , d) be a complete separable metric space, and denote by$X$ .
as the subset of
We also define
space of Borel probability measures on$\int_{X}d(x,y)^{2}\mu(dy)<\infty$all) $x\in X.$
(and
hence
for
some
if
such that$\mathcal{P}(X)$$\mathcal{P}(X)$$\mathcal{P}^{2}(X)$$\mu\in \mathcal{P}^{2}(X)$Definition 1.1 (Wasserstein spaces) For
of and is defined by$\mu$$\mu,$$\nu\in \mathcal{P}^{2}(X)$, the$L^{2}$-Wasserstein distance$\nu$$W_{2}( \mu, \nu) :=\inf_{\pi}(\int_{XxX}d(x, y)^{2}\pi(dxdy))^{1/2}$

whereruns over all couplings of andfor every Borel set $A\subset X$ . We call$\pi\in \mathcal{P}(X\cross X)$$\pi(X\cross A)=\nu(A)$

$\mu$

$v$

, i.e., $\pi(A\cross X)=\mu(A)$ and$(\mathcal{P}^{2}(X), W_{2})$the$L^{2}$-Wassersteinspace over $X.$

In view of optimal transport theory, $d(x, y)^{2}$ is the cost we pay for transporting theunit mass from to $\pi(x, y)$ represents the mass transported from to , and(i.e.,
minimal geodesic with respect tois the least cost for transporting towith $W_{2}(\mu_{s}, \mu_{t})=|s-t|W_{2}(\mu_{0}, \mu_{1})$ for all $t\in[0,1])$ describes antransport.
wayof
optimal
$x$

$x$

$y,$

$\mu$

$(\mu_{t})_{t\in[0,1]}\subset \mathcal{P}^{2}(X)$

$v.$

$A$

$y$

$W_{2}(\mu, v)^{2}$$W_{2}$$s,$

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (sohta@math.kyoto-u.ac.jp);
Supported in part by the Grant-in-Aid for Young Scientists (B) 23740048.
$*$

45
If we fix a Borel measure on $X$ , thenwhich is absolutely continuous with respect to$\mathcal{P}_{ac}(X, \omega)$$\omega$

will denote the set of$\mu\in \mathcal{P}(X)$$\omega.$

Definition 1.2 (Relative entropy) Define the relative entropy
with respect to by$Ent_{\omega}(\mu)$of$\mu\in \mathcal{P}(X)$$\omega$$Ent_{\omega}(\mu):=\int_{\sup p\rho}\rho\log\rho d\omega,$

provided that$\mu=\rho\omega\in \mathcal{P}_{ac}(X, \omega)$

and$\int_{\{\rho>1\}}\rho\log\rho d\omega<\infty$. Otherwise, set$Ent_{\omega}(\mu)$$:=$$\infty.$

Observe that, for a Borel set $A\subset X$ with $0<\omega(A)<\infty$ , the uniform distribution$\mu_{A}:=\omega(A)^{-1}\cdot\omega|_{A}$

1.2
Let
that

on

$A$

satisfies $Ent_{\omega}(\mu_{A})=-\log(\omega(A))$ .

McCann’s displacement convexity$L$be the Lebesgue measure on
is ‘nonnegatively curved’.

$\mathbb{R}^{n}$. McCann’s following pioneering theorem means$\mathbb{R}^{n}$

Theorem 1.3 (Convexity of $Ent_{L}$ ; [Mcl]) The relative entropy $Ent_{L}$ is convex $on$
in the sense that$(\mathcal{P}^{2}(\mathbb{R}^{n}), W_{2})$(1. 1)$Ent_{L}(\mu_{t})\leq(1-t)Ent_{L}(\mu_{0})+tEnt_{L}(\mu_{1})$for all $t\in[0,1]$ along any minimal geodesic$(\mu_{t})_{t\in[0,1]}\subset \mathcal{P}^{2}(\mathbb{R}^{n})$with respect to$W_{2}.$

We sometimes call (1.1) the displacement convexity in order to emphasis the difference
from the convexity along the convex combination $t\mapsto(1-t)\mu_{0}+t\mu_{1}.$One may understand the validity of the convexity of $Ent_{L}$ from the geometric viewpoint
as follows. The classical Brunn-Minkowski inequality on
asserts that, for any Borel
$A,$
sets
and $t\in[0,1],$$\mathbb{R}^{n}$$B\subset \mathbb{R}^{n}$$L((1-t)A+tB)^{1/n}\geq(1-t)L(A)^{1/n}+tL(B)^{1/n}$ .

(1.2)

Since the function $s\mapsto\log(s)$ is increasing and concave, (1.2) immediately implies$Ent_{L}(\mu_{(1-t)A+tB})\leq(1-t)Ent_{L}(\mu_{A})+tEnt_{L}(\mu_{B})$.

Hence (1.1) can be regarded as a weaker (dimension-free) form of the Brunn-Minkowski
inequality.

1.3

Ricci curvature and convexity of relative entropy

Let $(M, g)$ be a complete Riemannian manifold, and denote by
its Riemannian volume
measure. We always assume that $M$ is connected and boudaryless.
In their influential paper [OV], Otto and Villani gave a heuristic argument (to bemade rigorous in [LV2] on how a lower Ricci curvature bound implies several functional
inequalities via the convexity of the relative entropy. Their discussion was basedon formal calculus in terms of Otto’s Riemannian structure of([Ot]).
$\omega_{g}$$)$$Ent_{\omega_{g}}$$(\mathcal{P}^{2}(M), W_{2})$46
Then, Cordero-Erausquin et al [CMS] showed up with a rigorous connection between. We say that $Ric_{g}\geq K$
the lower Ricci curvature bound and the convexity ofdenotes the Ricci
if $Ric_{g}(v, v)\geq K|v|^{2}$ for all $v\in TM$ , where
holds for somecurvature tensor of$Ent_{\omega_{9}}$$Ric_{g}$$K\in \mathbb{R}$$g.$

Theorem 1.4 (Ricci bound implies convexity; [CMS]) Let $(M, g)$ be compact.
, i. e.,$Ric_{g}\geq K$
is $K$-convex on, thenfor some$K\in \mathbb{R}$(1.3)$Ent_{\omega_{9}}(\mu_{t})\leq(1-t)Ent_{\omega_{g}}(\mu_{0})+tEnt_{\omega_{g}}(\mu_{1})-\frac{K}{2}(1-t)tW_{2}(\mu_{0}, \mu_{1})^{2}$

holds for all $t\in[O, 1]$ along any minimal geodesic$(\mu_{t})_{t\in[0,1]}\subset \mathcal{P}(M)$We remark that, on a Riemannian manifold, any pair
by a unique minimal geodesic (see [Mc2]).

2

If$(\mathcal{P}(M), W_{2})$$Ent_{\omega_{g}}$$\mu,$.

$\nu\in \mathcal{P}_{ac}^{2}(M, \omega_{g})$is connected

Curvature-dimension condition

The converse implication of Theorem 1.4 also holds true.
and a
Theorem 2.1 (Ricci bound is equivalent to convexity; $[vRS]$ ) For$K$
is -convex.
complete Riemannian manifold $(M, g),$ $Ric_{g}\geq K$ holds if and only if$K\in \mathbb{R}$$Ent_{\omega_{9}}$is formulated without using the differentiable
Note that the $K$-convexity of$M$
structure of , we need only a distance and a measure. Thus Theorem 2.lled us to
the following notion of ‘metric measure spaces with lower Ricci curvature bounds’. Let$(X, d, \omega)$ be a complete, separable metric space equipped with a Borel measure
on $X$$U\subset
X.$such that $0<\omega(U)<\infty$ for any nonempty, bounded open set$Ent_{\omega_{g}}$$\omega$, we sayDefinition 2.2 (Curvature-dimension condition; [Stl], [LV2]) For$CD(K,\infty)$if any pairthat $(X, d, \omega)$ satisfies the curvature-dimension condition(1.3) holds for
which
along
geodesic
is connected by a minimal
all $t\in[0,1].$$K\in \mathbb{R}$$\mu_{0},$$\mu_{1}\in$$(\mu_{t})_{t\in[0,1]}\subset \mathcal{P}^{2}(X)$$\mathcal{P}^{2}(X)$

Remark 2.3 (a) In general, minimal geodesics in the Wasserstein space are not unique$(e.g., over the$ normed space).
reason why we impose (1.3) only along someminimal geodesic is to make this condition stable under the convergence of underlying
spaces (see Theorem 2.4 below).
(b) The word “curvature-dimension condition” comes from Bakry and \’Emery’s celebrated theory on linear semigroups and functional inequalities (see [BE]). We can introand $N\in(1, \infty] (but the$ definition $is more$ involved) ,duce $CD(K, N)$ for generalsatisfies $CD(K, N)$ if and only if $Ric_{g}\geq K$ andand a Riemannian manifold$\dim M\leq N$ ([St2], [LVl]).
(c) If $(X, d, \omega)$ satisfies $CD(K, N)$ , then it also satisfies $CD(K’, N’)$ for any $K’<K$
and $N’>N.$$(\mathbb{R}^{n}, |\cdot|_{\infty})$$A$$K\in \mathbb{R}$$(M, g, \omega_{g})$There are a number of geometric and analytic applications of $CD(K, N)$ , including(see [St2], [LVl], [LV2] for details):

47$\bullet$$\bullet$$\bullet$Bishop-Gromov volume comparison theorem for $N<\infty$ ;Talagrand, $log$ -Sobolev, and global Poincare inequalities for $K>0$ and $N=\infty$ ;Bonnet-Myers diameter bound and Lichnerowicz inequality for $K>0$ and $N<\infty.$

Another, geometric motivation behind the study of $CD$ stems from the Gromov-Fukaya
pre-compactness ([Gr], [Fu]), which asserts that a sequenceof complete Riemannian manifolds with uniform bounds $Ric_{g_{i}}\geq K$ and $\dim M_{i}\leq N<\infty$ contains a
subsequence convergent in the sense of the (pointed) measured Gromov-Hausdorff convergence. Such a limit space is not necessarily a Riemannian manifold any more, but
still possesses nice properties as was established in Cheeger and Colding’s series of works([CC]). By the following stability theorem, limit spaces certainly satisfy $CD(K, N)$ .
$\{(M_{i}, g_{i})\}_{i\in \mathbb{N}}$Theorem 2.4 (Stability; [Stl], [LV2]) Suppose that a sequence of metric measurespacesconverges to a metric measure space $(X, d, \omega)$ in the sense of the(pointed) measured Gromov-Hausdorff convergence. If
satisfies $CD(K, N)$ for$N\in(1, \infty] and all i, then (X, d, \omega)$ also satisfies $CD(K, N)$ .
some$\{(X_{i}, d_{i}, \omega_{i})\}_{i\in N}$$(X_{i}, d_{i}, \omega_{i})$$K\in \mathbb{R},$However, it turned out that $CD$ covers a much wider class than the closure of Riemannian manifolds. A Finsler manifold $(M, F)$ is a generalization of a Riemannian manifold
such that $F:TMarrow[O, \infty)$ gives (sufficiently smooth, convex) norm on each tangent
space $T_{x}M$ (see [BCS] for more details).
$a$

Theorem 2.5 (Finsler case; $[Oh2]$ ) Let $(M, F)$ be forward complete Finsler mani-measure on M. Then $Ric_{N}\geq K$ holds if and only iffold equipped with a positive$(M, d_{F}, \omega)$$CD(K,
N)$, where $Ric_{N}$ is the weighted Ricci curvature with respect tosatisfies, andis the distance function induced from $F.$$a$

$C^{\infty}$$\omega$$\omega$$d_{F}$

In particular, any normed spacesatisfies $CD(O, n)$ , whereas normed spaces(other than inner product spaces) can not appear as the limit of Riemannian manifolds.
On the one hand, this is a good news since $CD$ is available in a wide class of spaces.
On the other hand, one can expect a stronger condition than $CD$ that rules out normed
spaces and has stronger consequences. The key ingredient to answer this question is the
behavior of heat flow.
$(\mathbb{R}^{n}, |\cdot|, L)$

3

Heat flow as gradient flow

We can introduce heat flow as gradient flow in the following two ways:(I) the gradient flow of the energy$\mathcal{E}$in the$L^{2}$-space;(II) the gradient flow of the relative entropy in the$L^{2}$-Wasserstein space.

The first approach (I) is classical. On a general metric measure space $(X, d, \omega),$
introduced as the Cheeger energy:$\mathcal{E}(u):=\frac{1}{2}\int_{X}|\nabla u|^{2}d\omega, |\nabla u|(x):=\lim_{yarrow}\sup_{x}\frac{|u(x)-u(y)|}{d(x,y)},$$\mathcal{E}$is48
for a locally Lipschitz function $u:Xarrow \mathbb{R}$ (see [Ch], [AGSI] for the precise definition).
The second, much newer approach (II) was initiated by Jordan, Kinderlehrer and$Otto^{)}s$ seminal work ([JKO]). The identification (I)(II) was then extended to Riemannian manifolds ([Ohl], [Sa], [Er]), Finsler manifolds ([OSl]), Alexandrov spaces ([GKO]),and finally to general metric measure spaces satisfying $CD$ ([AGSI]).
On a non-Riemannian Finsler manifold, however, heat flow is nonlinear. The nonlinearity causes several difficulties in applications. Then, it would be worthwhile to consider
spaces satisfying $CD$ such that heat flow is linear, that turned out a nice condition.
$=$

4

Riemannian curvature-dimension condition

A metric measure space $(X, d, \omega)$ is said to satisfy strong $CD(K, \infty)$ if (1.3) holds alongsatisfies. For instance,inevery minimal geodesic$CD(O, n)$ but does not satisfy strong $CD(0, \infty)$ .
$(\mu_{t})_{t\in[0,1]}$$(\mathbb{R}^{n}, |\cdot|_{\infty},L)$$(\mathcal{P}^{2}(X), W_{2})$Definition 4.1 (Riemannian curvature-dimension condition; [AGS2]) We saythat a metric measure space $(X, d, \omega)$ satisfies the Riemannian curvature-dimension condition RCD $(K, \infty)$ if it satisfies strong $CD(K, \infty)$ and the heat flow on it is linear.

Similarly to $CD$ , RCD $(K, \infty)$ is preserved under the (pointed) measured Gromovwith $Ric_{g}\geq K$
Hausdorff convergence ([AGS2]). Since Riemannian manifolds
satisfy RCD $(K, \infty)$ , their limit spaces also satisfy RCD $(K, \infty)$ . The following characterizations of RCD $(K, \infty)$ are useful and inspiring.
$(M, g, \omega_{g})$Theorem 4.2 (Equivalent conditions to RCD; [AGS2]) The condition RCD $(K, \infty)$
is equivalent to:(i) strong $CD(K, \infty)$ and that$\mathcal{E}$is quadratic (so that$\mathcal{E}$induces a Dirichlet form);(ii) the evolution variational inequality:$\frac{d}{dt}[\frac{W_{2}(\mu_{t},\nu)^{2}}{2}]+\frac{K}{2}W_{2}(\mu_{t}, \nu)^{2}+Ent_{\omega}(\mu_{t})\leq Ent_{\omega}(\nu)$for all$(\mu_{t})_{t>0}\subset \mathcal{P}^{2}(X)$obeying heat flow, all$\nu\in \mathcal{P}^{2}(X)$, and for $a.e.$ $t>0.$

Roughly speaking, the evolution variational inequality is derived by estimating the firstalong a minimal geodesic between
by the $K$ -convexity ofvariationand , for which we essentially need the Riemannian structure’ (i.e., angles). In asimilar manner, we can control the distance between two curves obeying heat flow.
$Ent_{\omega}$$\frac{d}{dt}[W_{2}(\mu_{t}, \nu)^{2}/2]$$($$\mu_{t}$$\nu$Theorem 4.3 ( $K$ -contraction property) Let us assume RCD $(K, \infty)$ . Then, for any
along heat flow, we have
curves$(\mu_{t})_{t\geq 0},$$(\nu_{t})_{t\geq 0}\subset \mathcal{P}^{2}(X)$$W_{2}(\mu_{t}, \nu_{t})\leq e^{-Kt}W_{2}(\mu_{0}, v_{0}) \forall t>0.$This is a standard consequence on the gradient flow of a $K$-convex function, while weneed the ‘Riemannian structure’ for the same reason as the evolution variational inequality. Thanks to Kuwada’s duality ([Ku]) on linear semigroups, the $K$ -contraction property
implies the following.

49
Theorem 4.4 (Bakry-Emery type gradient estimate) Suppose RCD $(K, \infty)$ and let
be along heat flow with. Then we have$(\rho_{t})_{t\geq 0}$$\rho_{t}\omega\in \mathcal{P}^{2}(X)$$|\nabla\rho_{t}|(x)^{2}\leq e^{-2Kt}P_{t}(|\nabla\rho_{0}|^{2})(x) \forall x\inX, \forall t>0,$

where$P_{t}$

is the heat semigroup$(i.e., P_{t}(|\nabla\rho_{0}|^{2})$is the heat flow starting from$|\nabla\rho_{0}|^{2}$).

Furthermore, we obtain the (dimension-free) Bochner inequality$\frac{1}{2}\triangle(|\nabla u|^{2})-D(\triangle u)(\nabla u)\geq K|\nabla u|^{2}$in a certain weak sense ([GKO], [AGS2]).

5
5.1

Further problems$CD$vs Finsler

Recall that a Finsler manifold satisfies $CD$ but does not satisfy RCD unless it is a Riemannian manifold. Therefore Finsler manifolds are reasonable model spaces satisfying $CD.$
The following theorem reveals the difference between Riemannian and Finsler manifolds(in other words, the difference between RCD- and $CD$ -spaces).
Theorem 5.1 (Non-contraction of heat flow; [OS2]) The heat flow of a normed
space
is not $K$ -contractive for any
unless the norm
comes from an$(\mathbb{R}^{n}, |\cdot|, L)$$K\in \mathbb{R}$$|\cdot|$

inner product.

We can nevertheless show the Bochner- Weitzenbock formula on general Finsler manifolds, and as applications the (modified versions of) Bakry-Emery type and Li-Yau typegradient estimates as well as the Harnack inequality ([OS3]). Furthermore, the CheegerGromoll type (homeomorphic) splitting theorem was generalized in [Oh4].
For general $CD$ -spaces, Gigli recently showed the Laplacian comparison theorem ([Gi]).
The Bochner-Weitzenb\"ock formula and the gradient estimates are not known on general$CD$ -spaces.

5.2

RCD $(K, N)$ for $N<\infty$ ?

It is unclear how to define RCD $(K, N)$ for $N<\infty$ (especially with $K\neq 0$ ). RCD $(K, \infty)$
is naturally related to the behavior of heat flow via the relative entropy. $CD(K, N)$ is
somehow related to the fast diffusion equation, whereas it is always
nonlinear. Moreover, even in the Riemannian setting, it is only partially known about thecontraction property and gradient estimates corresponding to $CD(K, N)$ . An appropriate
notion of RCD $(K, N)$ should imply, for instance, the Li-Yau type gradient estimate.
$\partial_{t}u=\triangle(u^{(N-1)/N})$

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