Extended Version On the Geometry of etric easure Spaces. II. Karl-Theodor Sturm This is a continuation of our previous paper [St04] On the Geometry of etric easure Spaces where we introduced and analyzed lower ( Ricci ) curvature bounds Curv K for metric measure spaces (, d, m). The definition used there is based on convexity properties of the relative entropy Ent(. m) regarded as a function on P 2 (, d), the L 2 -Wasserstein space of probability measures on the metric space (, d). For Riemannian manifolds, Curv(, d, m) K if and only if Ric (ξ, ξ) K ξ 2 for all ξ T. This notion of lower curvature bound is a dimension independent (or, in a certain sense, infinite dimensional ) concept. In order to obtain more precise estimates one has to reinforce the curvature bound Curv K to a curvature-dimension condition CD(K, N) involving two parameters K and N playing in some generalized sense the roles of a lower bound for the Ricci curvature and an upper bound for the dimension, resp. The main topic of the present paper is the curvature-dimension condition CD(K, N) for metricmeasure spaces (, d, m). In some sense, it will be the geometric counterpart to the curvaturedimension condition for arkov operators and Dirichlet forms by Bakry and Émery [BE85]. We will also study a weak variant of the latter, namely the measure contraction property CP(K, N). It is a slight modification of a property introduced in [St98] and in a similar form in [KS03]. As in [St04], our definition of the curvature-dimension condition is based on a kind of convexity property for suitable functionals on the L 2 -Wasserstein space P 2 (, d). The previous curvature condition Curv K is included as the limit case CD(K, ). For finite N the basic object now is the Rényi entropy functional S N (ρm m) = ρ /N dm, replacing the relative ( Shannon ) entropy Ent(ρm m) = ρ log ρ dm = lim N( + S N(ρm m)). N For Riemannian manifolds, the curvature-dimension condition CD(K, N) will be satisfied if and only if dim() N and Ric (ξ, ξ) K ξ 2 for all ξ T. Under minimal regularity assumptions, condition CD(K, N) will imply property CP(K, N); the latter will be strictly weaker. Roughly spoken, CD(K, N) is a condition on the optimal transport between any pair of (absolutely continuous) probability measures on whereas CP(K, N) is a condition on the optimal transport between Dirac masses and the uniform distribution on. One of the fundamental results is that the curvature-dimension condition as well as the measure contraction property are stable under convergence w.r.t. the distance D. The latter was introduced in [St04] as a complete separable metric on the family of (isomorphism classes of) See also [St05a] for a comprehensive version.
normalized metric measure spaces. oreover, we deduce that for each triple (K, N, L) R 3, the family of normalized metric measure spaces (, d, m) which have diameter L and which satisfy condition CD(K, N) (or alternatively, property CP(K, N)) is compact. Furthermore, we present various geometric consequences of the curvature-dimension condition (or alternatively of the measure contraction property). The most prominent among them being the Bishop-Gromov theorem on the volume growth of concentric balls and the Bonnet-yers theorem on the diameter of metric measure spaces with positive lower curvature bounds. In both cases, we obtain the sharp estimates known from the Riemannian case. Of particular interest are the analytic consequences of property CP(K, N). It allows to construct a canonical Dirichlet form and a canonical Laplace operator on L 2 (, m), it implies a local Poincaré inequality, a scale invariant Harnack inequality, and Gaussian estimates for heat kernel and it yields Hölder continuity of harmonic functions. The curvature-dimension condition can be interpreted as a control on the distortion of infinitesimal volume elements under transport along geodesics. Let us briefly try to explain this. The curvature-dimension condition CD(0, N) which besides the condition CD(K, ) is the easiest to formulate simply states that for all N N the functional S N (. m) is convex on the L 2 -Wasserstein space P 2 (, d). In the Riemannian case, it was already observed in [St05] that the latter characterizes manifolds with dimension N and Ricci curvature 0. This is basically due to the fact that the Jacobian determinant J t = det df t of any transport map F t := exp( t ϕ) : satisfies 2 t 2 J /N t (x) 0 (0.) if and only if has dimension N and Ricci curvature 0. Essentially equivalent to (0.) is the Brunn-inkowski inequality: m(a t ) /N ( t) m(a 0 ) /N + t m(a ) /N (0.2) for any N N, any t [0, ], and any pair of sets A 0, A where A t denotes the set of point γ t on geodesics with endpoints γ 0 A 0, γ A. The curvature-dimension condition CD(K, N) for general K and N is more involved. As a first step, the inequality (0.) can be replaced by 2 t 2 J /N t (x) K N J /N t (x) d 2 (x, F (x)) (see [St05], Corollary 3.4). A more refined analysis yields J /N t (x) τ ( t) ( d(x, F (x))) J /N 0 (x) + τ (t) ( d(x, F (x))) J /N (x) (0.3) for t [0, ] and x where with κ = K N τ (t) (θ) = t/n ( ) sin (κ tθ) /N sin (κ θ) (and with appropriate interpretation if K 0). The curvature-dimension condition CD(K, N) to be discussed in the sequel can be regarded as a robust version of (0.3). Assume for simplicity that for m m-a.e. (x, y) 2 there exists a unique geodesic t γ t (x, y) depending in a measurable way on the endpoints x and y. Then CD(K, N) states that that for 2
any pair of absolutely continuous probability measures ρ 0 m and ρ m on there exists an optimal coupling q such that [ ρ t (γ t (x, y)) τ ( t) ( d(x, y)) ρ /N 0 (x) + τ (t) ( d(x, y)) ρ /N (y)] N (0.4) for all t [0, ], and q-a.e. (x, y) 2 where ρ t is the density of the push forward of q under the map (x, y) γ t (x, y). Roughly spoken, CP(K, N) is the particular case where ρ 0 m is degenerated to a Dirac mass. This amounts to say that for every x and t [0, ] the contracted measure m t,x := γ t (x,.) m satisfies [ ] sin (κ d(x, y)) N t m t,x(dy) m(dy). (0.5) sin (κ d(x, y)/t) In the general case, the assumption of a measurable choice of a unique geodesic is replaced by the assumption of a measurable choice of a measure on the geodesics. Independently of [St05] and [St04], the particular cases CD(K, ) and CD(0, N) of the curvaturedimension condition defined in essentially the same form were also discussed in a recent paper [LV04] by John Lott and Cédric Villani. In the first chapter of this paper, we introduce the curvature-dimension condition and we deduce some of the basic properties. In the second chapter, we derive various geometric consequences of the curvature-dimension condition like Brunn-inkowski inequality, Bishop-Gromov volume growth estimate, and Bonnet- yers theorem. The topic of chapter 3 is the stability of the curvature-dimension condition under convergence. oreover, compactness of families of normalized metric measure spaces with suitable bounds on the diameter, the dimension and the curvature is deduced. In chapter 4 we study the curvature-dimension condition under the additional assumption that the underlying space is nonbranching. In chapter 5 we introduce and study the measure contraction property, including its geometric consequences, and its stability under convergence. Chapter 6 is devoted to the analytic consequences of the measure contraction property, in particular, the construction of Sobolev spaces and Dirichlet forms as well as the derivation of a scale invariant local Poincaré inequality. Throughout this paper, we freely use definitions and results from our previous paper [St04]. Theorem x.y and equation (a.b) of that paper will be quoted as Theorem I.x.y or (I.a.b), resp. The Curvature-Dimension Condition A metric measure space will always be a triple (, d, m) where (, d) is a complete separable metric space and m is a locally finite measure (i.e. m(b r (x)) < for all x and all sufficiently small r > 0) on equipped with its Borel σ-algebra. To avoid pathologies, we exclude the case m() = 0. G() will denote the space of geodesics γ : [0, ], equipped with the topology of uniform convergence. Here and in the sequel by definition each geodesic is minimizing and parametrized proportional to arclength. A point z will be called t-intermediate point of points x and y if d(x, z) = t d(x, y) and d(z, y) = ( t) d(x, y). 3
P 2 (, d) denotes the L 2 -Wasserstein space of probability measures on and d W the corresponding L 2 -Wasserstein distance. The subspace of m-absolutely continuous measures is denoted by P 2 (, d, m). X denotes the family of all isomorphism classes of metric measure spaces and X the subfamily of isomorphism classes of normalized metric measure spaces (, d, m) with finite variances (i.e. m() = and d2 (o, y) dm(y) < ). On X we have introduced the distance D, see chapter 3 in [St04]. Given a metric measure space (, d, m) and a number N R, N we define the Rényi entropy functional S N (. m) : P 2 (, d) R with respect to m by S N (ν m) := ρ /N dν where ρ denotes the density of the absolutely continuous part ν c in the Lebesgue decomposition ν = ν c + ν s = ρm + ν s of ν P 2 (, d). Note that in the borderline case N = this reads S (ν m) := m(supp[ν c ]). Instead of S N, mostly in the literature the functional S N := N+N S N is considered.the latter shares various properties with the relative Shannon entropy Ent(. m). For instance, if m is a probability measure then S N (. m) 0 on P 2 (, d) and S N (ν m) = 0 if and only if ν = m. For the purpose of this paper, the functional S N from above is more convenient. We recall two important facts from the proof of Lemma I.4.. Lemma.. Assume that m() is finite. (i) Then for each N > the Rényi entropy functional S N (. m) is lower semicontinuous and satisfies m() /N S N (. m) 0 on P 2 (, d). (ii) For each ν P 2 (, d) Ent(ν m) = lim N N( + S N(ν m)). Given two numbers K, N R with N we put for (t, θ) [0, ] R + :, if Kθ 2 (N )π 2 ( ) ( )) /N t /N K (sin τ (t) N tθ K /sin N θ, if 0 < Kθ 2 < (N )π 2 (θ) := t, if Kθ 2 = 0 or if Kθ 2 < 0 and N = ( ) ( )) /N t (sinh /N K N /sinh tθ K N θ, if Kθ 2 < 0 and N >. That is, τ (t) (θ) := t/n σ (t) (θ) /N ( ) where σ (t) (θ) := sin K N /sin tθ 0 < Kθ 2 < Nπ 2 and with appropriate interpretation otherwise. oreover, we put ( K N θ ) if ς (t) (t) (θ) := τ (θ)n. Straightforward calculations yield that for fixed t ]0, [ and θ ]0, [ the function (K, N) τ (t) (θ) is continuous, nondecreasing in K and nonincreasing in N. oreover, 4
Lemma.2. For all K, K R, all N, N ]0, [, all t [0, ] and all θ R + : and, if N, σ (t) (θ)n σ (t) K,N (θ) N σ (t) K+K,N+N (θ) N+N τ (t) (θ)n σ (t) K,N (θ) N τ (t) K+K,N+N (θ) N+N. Proof. We derive the first inequality; the rest follows easily. For each fixed t ]0, [ the function f : K log sin( Kt) sin( (with canonical interpretation for nonpositive K) is convex on ], K) π2 [. Hence, for all K, K R, all N, N > 0 and all θ R under consideration N N + N f ( K N θ2 ) + N ( K N + N f N θ2 ) ( ) K + K f N + N θ2. In other words, ( K sin sin ) N tθ ( ) K N θ N sin ( K sin ) N tθ ) ( K N θ N ( K+K sin sin N+N tθ ( K+K N+N θ ) ) N+N. In particular, τ (t) (θ) σ(t) (θ) provided N > since τ N = σ 0, σn. Definition.3. Given two numbers K, N R with N we say that a metric measure space (, d, m) satisfies the curvature-dimension condition CD(K, N) iff for each pair ν 0, ν P 2 (, d, m) there exist an optimal coupling q of ν 0, ν and a geodesic Γ : [0, ] P 2 (, d, m) connecting ν 0, ν with [ S N (Γ(t) m) τ ( t) ( d(x 0, x )) ρ /N 0 (x 0 ) ] + τ (t) ( d(x 0, x )) ρ /N (x ) dq(x 0, x ) (.) for all t [0, ] and all N N. Here ρ i denotes the density of the absolutely continuous part of ν i w.r.t. m (for i = 0, ). The definition of the curvature-dimension condition immediately implies its invariance under standard transformations of metric measure spaces. (Cf. also Propositions I.4.2, I.4.3, and I.4.5 as well as the proofs of these results.) Proposition.4. Let (, d, m) be a metric measure spaces which satisfies the CD(K, N) condition for some pair of real numbers K, N. Then the following properties hold: (i) Isomorphism : Each metric measure space (, d, m ) which is isomorphic to (, d, m) satisfies the CD(K, N) condition. (ii) Scaled spaces : For each α, β > 0 the metric measure space (, α d, βm) satisfies the CD(α 2 K, N) condition. (iii) Subsets : For each convex subset of the metric measure space (, d, m) satisfies the same CD(K, N) condition. 5
Remark.5. In order that the curvature-dimension condition is invariant under isomorphisms we require (.) to hold true only for ν 0, ν P 2 (, d, m) and not for all ν 0, ν P 2 (, d). For instance, consider = R n equipped with the Euclidean distance d and let m be the (n )- dimensional Lebesgue measure on 0 := {0} R n. Then (, d, m) satisfies the condition CD(0, n ). However, choose ν 0 = A m for some set A 0 with m(a) = and ν = δ z for some point z = (z,..., z n ) with z 0. Then for each midpoint Γ /2 of them Proposition.6. 0 = S n (Γ /2 m) 2 S n (ν 0 m) + 2 S n (ν m) = 2. (i) If (, d, m) satisfies the curvature-dimension condition CD(K, N) then it also satisfies the curvature-dimension conditions CD(K, N ) for all K K and N N. Conversely, if (, d, m) is compact with diameter L and satisfies the curvature-dimension conditions CD(K n, N n ) for a sequence of pairs (K n, N n ) with lim n (K n, N n ) = (K, N) and K L 2 < (N )π 2 then it also satisfies the curvature-dimension conditions CD(K, N). (ii) If (, d, m) has finite mass and satisfies the curvature-dimension condition CD(K, N) for some K and N then it has curvature K in the sense of Definition I.4.5. Therefore, the condition Curv(, d, m) K may be interpreted as the curvature-dimension condition CD(K, ) for (, d, m). (iii) A metric measure space (, d, m) satisfies the curvature-dimension condition CD(0, N) for some N if and only if the Rényi entropy functionals S N (. m) for N N are weakly convex on P 2 (, d, m) in the following sense: for each pair ν 0, ν P 2 (, d, m) there exist a geodesic Γ : [0, ] P 2 (, d, m) connecting ν 0, ν with for all t [0, ] and N N. S N (Γ(t) m) ( t) S N (ν 0 m) + t S N (ν m) (.2) Proof. (i): The first assertion is obvious. The second one will follow from Theorem 3. below. (ii): Let ν 0, ν P 2 (, d, ) be given with Ent(ν 0 m) < and Ent(ν m) <. By assumption, (, d, m) satisfies the curvature-dimension condition CD(K, N) for some K and N. Hence, there exists there exist an optimal coupling q of ν 0, ν and a geodesic Γ : [0, ] P 2 (, d, m) connecting ν 0, ν with (.) for all t [0, ] and all N N. The assumption m() < implies that Ent(Γ t m) = lim N N ( + S N (Γ t m)) for all t [0, ]. Hence, (iii): Obvious. Ent(Γ t m) ( t)ent(γ 0 m) tent(γ m) = lim N (S N (Γ t m) ( t)s N (Γ 0 m) ts N (Γ m)) N ( [ ] lim N ( t) τ ( t) N ( d(x 0, x )) ρ /N 0 (x 0 ) [ ] ) + N t τ (t) ( d(x 0, x )) ρ /N (x ) dq(x 0, x ) ( [ ] lim N ( t) τ ( t) N ( d(x 0, x )) [ ]) + N t τ (t) ( d(x 0, x )) dq(x 0, x ) t( t) = K d 2 t( t) (x 0, x )dq(x 0, x ) = K d 2 W (ν 0, ν ). 2 2 6
Proposition.6(iii) gives an elementary characterization of the condition CD(0, N) through convexity of the Rényi entropy functionals S N for N N. A generalization of this characterization to K 0 will be discussed in chapter 4. oreover, we will present various modifications of the curvature-dimension condition which formally are more restrictive. Of course, the most important case to be studied is the case of Riemannian manifolds. Let us mention here the basic result. We postpone its proof to the end of the chapter. Theorem.7. Let be a complete Riemannian manifold with Riemannian distance d and Riemannian volume m and let numbers K, N R with N be given. (i) The metric measure space (, d, m) satisfies the curvature-dimension condition CD(K, N) if and only if the Riemannian manifold has Ricci curvature K and dimension N. (ii) oreover, in this case for every measurable function V : R the weighted space (, d, V m) satisfies the curvature-dimension condition CD(K + K, N + N ) provided Hess V /N K N V /N for some numbers K R, N > 0 in the sense that V (γ t ) /N σ ( t) K,N ( d(γ 0, γ )) V (γ 0 ) /N + σ (t) K,N ( d(γ 0, γ )) V (γ ) /N (.3) for each geodesic γ : [0, ] and each t [0, ]. Let us have a closer look on these results if is a subset of the real line equipped with the usual distance d and the -dimensional Lebesgue measure m. Example.8. (i) For each pair of real numbers K > 0, N > the space ([0, L], d, V m) with N L := K π and ( ) N K V (x) = sin N x satisfies the curvature-dimension condition CD(K, N). (ii) For each pair of real numbers K 0, N > the space (R +, d, V m) with ( ) N K V (x) = sinh N x, if K < 0, and V (x) = x N, if K = 0, satisfies the curvature-dimension condition CD(K, N). (iii) For each pair of real numbers K < 0, N > the space (R, d, V m) with ( ) N K V (x) = cosh N x satisfies the curvature-dimension condition CD(K, N). Note that for N the weight V from example (iii) from above converges to the weight ( ) K V (x) = exp 2 x2 from example I.4.0.(ii). Also note that according to [BQ00], the examples (i)-(iii) equipped with natural weighted Laplacians are also the prototypes for the Bakry-Emery curvature-dimension condition. 7
Proof of Theorem.7. (a) Let be a complete Riemannian manifold with Ricci curvature K and dimension n N and assume that we are given two absolutely continuous probability measures ν 0 = ρ 0 m and ν = ρ m in P 2 (, d, m). Without restriction, we may assume that both are compactly supported. (Otherwise, we have to choose compact exhaustions of and to consider the restriction of the coupling to these compact sets). According to Remark I.2.2(iii), there exists a weakly differentiable function ϕ : R such that the push forward measures with Γ t = (F t ) ν 0 F t (x) = exp x ( t ϕ(x)) for t [0, ] define the unique geodesic t Γ t in P 2 (, d) connecting ν 0 and ν. Again each Γ t is compactly supported and absolutely continuous, say Γ t = ρ t m. Following [CS0] we may choose ϕ in such a way that it is d 2 /2-concave 2 and such that for ν 0 -a.e. x the Hessian of ϕ at x exists and the Jacobian df t (x) is nonsingular for all t [0, ]. (b) For each x and t as above, consider the matrix of Jacobi fields A t (x) := df t (x) : T x T Ft(x) along the geodesic F (x). (ore precisely, A t (x)v is a Jacobi field along F (x) for each v T x.) It is the unique solution of the Jacobi equation ( t t A t (x) + R A t (x), F t (x)) F t (x) = 0 with initial conditions A 0 = Id, t A t {t=0} = Hess ϕ. Here R is the curvature tensor and t denotes covariant derivates along the geodesics F (x) (cf. [Ch93] (3.4)). By assumption, the matrix A t (x) is non-degenerate for all x, t under consideration. Hence, the Jacobi equation immediately implies that the self-adjoint matrix valued map U t := t A t A t solves the Riccati type equation ( t U t + Ut 2 + R, F ) t F t = 0 (.4) and thus tr( t U t ) + tr(u 2 t ) + Ric ( F t, F t ) = 0. (.5) Now consider y t := log J t = log det A t. Then tru t = tr( t A t A t ) = d dt (log det A t) = ẏ t (cf. [Ch93], Prop. 2.8). Hence, tr( t U t ) = d dt tr(u t) = ÿ t. By means of the standard estimate tr(ut 2 ) n (tru t) 2 for the trace of the square of a self-adjoint matrix, we obtain from (.5) ÿ t nẏ2 ( t Ric F t, F ) t. (.6) Using our estimates for the dimension and the Ricci curvature of we get ÿ t N ẏ2 t Kθ 2 with θ(x) := F t (x) = d(x, F (x)) or equivalently d 2 dt 2 J /N t Kθ2 N J /N t. (.7) 2 This notion of concavity is defined in terms of some generalized Legendre transform. It is completely different from the notion of convexity/concavity along geodesics used at all other instances in this paper. 8
Integrating (.7) for fixed x along the geodesic t F t (x) yields J /N t σ ( t) /N (θ) J0 + σ (t) /N (θ) J. (.8) This is close to the estimate (.2) which we aim for. In the case n = we are already done. (c) In order to improve upon (.8) in the case n 2, we will separately study the deformation of the volume element in directions parallel and orthogonal to the transport direction. To be precise, fix x as above and let e t,..., e n t be an orthonormal basis of T Ft (x) with e t = / F t (x) F t (x) for all t [0, [. Put u ij (t) = e i t, U t e j t, λ t = + t 0 u (s)ds and L t = exp(λ t ). Then (.4) implies d n dt u (t) = u 2 j(t) u 2 (t). (.9) That is, λ t λ 2 t or, equivalently, L t 0 which in integrated form reads j= L t ( t)l 0 + tl (.0) for all t [0, ]. Now put α t = y t λ t, A t = exp(α t ) = J t /L t and V t = (u ij (t)) i,j=2,...,n. Then (.4) together with (.9) imply Hence, and thus A /(N ) t ( α t K θ 2 ÿ t Ric F t, F ) t + λ t = tr(ut 2 ) + λ n t = u 2 ij(t) d 2 ( dt 2 i,j= n u 2 ij(t) = tr(vt 2 ) i,j=2 n u 2 j(t) j= n (trv t) 2 = n α2 t N α2 t. A /(N ) t ) Kθ2 N A/(N ) t σ (( t), θ) A /(N ) 0 + σ (t, θ) A /(N ). (.) Finally, (.) and (.0) together with Hölder s inequality yield J /N t = (L t A t ) /N (( t)l 0 + tl ) /N (( t)l 0 ) /N + (tl ) /N ( σ ( t) ( σ (t) ( σ ( t) (θ) A/(N ) 0 + σ (t) (θ) A/(N ) 0 (θ) A/(N ) = τ ( t) /N (θ) J0 + τ (t) /N (θ) Jt. ) (N )/N ) (N )/N That is, the Jacobian determinant J t (x) := det df t (x) satisfies (θ) A/(N ) ) (N )/N J t (x) /N τ ( t) ( d(x, F (x)) J 0 (x) /N + τ (t) ( d(x, F (x)) J (x) /N (.2) 9
This estimate is from the technical point of view the main result in [CS0] (Lemma 6. and Cor. 2.2. To be precise, it is stated there only for the case N = n. However, the extension to the general case N n is straightforward.) For the convenience of the reader, we have presented here an alternative, essential self-contained derivation, following similar calculations in [St05]. (d) The change of variable formula for F t yields that ρ t (F t ) J t = ρ 0 a.e. Thus together with (.2) and (.3) we obtain (ρt ) N+N S N+N (Γ t V m) = N+N ρ t dm = J t V (F t ) N+N ρ N+N 0 dm V ( ) τ ( t) /N ( d) J0 + τ (t) N /N N+N ( d) J ( ) = ( σ ( t) K,N ( d) V (F 0 ) /N + σ (t) ( τ ( t) ( d) N N+N σ ( t) K,N ( d) N K,N ( d) V (F ) N+N + τ (t) ( d) N N+N σ (t) K,N ( d) N ( (τ ( t) K+K,N+N ( d) (J 0 V (F 0 )) N+N + τ (t) K+K,N+N ( d) (J V (F )) ( ( (τ ( t) ρ0 K+K,N+N ( d) + τ (t) K+K,N+N ( d) V ( ρ V ) N+N N+N J0 V (F 0 ) N+N J V (F ) N+N ) N+N ) N+N ) dq ρ N+N 0 dm N /N ) N+N ρ N+N 0 dm N+N ) N+N ρ N+N 0 dm which proves the claim. Here ( ) is due to Lemma.2. For the final equality we have used the fact that the unique optimal coupling of ν 0 and ν is given by dq(x 0, x ) = δ F (x 0 )(dx )dν(x 0 ). To simplify the above formulae, we always have dropped the arguments x 0 and x. To be more specific, d will denote d(x 0, F (x 0 )) if we integrate with respect to dm(x 0 ) and it will denote d(x 0, x ) if we integrate with respect to dq(x 0, x ). (e) Necessity: Assume that (, d, m) satisfies the curvature-dimension condition CD(K, N) for some pair of real numbers K, N with N. Then according to Corollary 2.5, N Hausdorff dimension of. In the Riemannian case, the latter coincides with the dimension. In order to prove that K Ricci curvature of, let us first investigate the case n 2. Assume the contrary: Ric z (ξ, ξ) (K δ) ξ 2 for some δ > 0, some point z and some ξ T z (). Consider the sets A 0 = B ɛ (exp z ( rξ)) and A = B ɛ (exp z (+rξ)). Then for sufficiently small ɛ r, Riemannian calculation yields m(a /2 ) ( + K δ r 2 + O(r 4 )) m(a 0) + m(a ) 2 2 (cf. [St05], Thm. 5.2) whereas Proposition 2. gives m(a /2 ) ( + K 2 r2 + O(r 4 )) m(a 0) + m(a ). 2 Hence, Ric K. Now let us investigate the case n =. Here the Ricci curvature always vanishes. On the other hand, n = implies N = (according to Corollary 2.5) and thus necessarily K 0 (according to Corollary 2.6, the generalized Bonnet-yers theorem). Hence, also in this case Ric K. 0
2 Geometric Consequences of the Curvature-Dimension Condition Compared with our defining property (.), the following version of the Brunn-inkowski inequality will be a very weak statement. However, it will still be strong enough to imply all the geometric consequences which we formulate in the sequel. Proposition 2. ( Generalized Brunn-inkowski Inequality ). Assume that the metric measure space (, d, m) satisfies the curvature-dimension condition CD(K, N) for some real numbers K, N R, N. Then for all measurable sets A 0, A with m(a 0 ) m(a ) > 0, all t [0, ] and all N N m(a t ) /N τ ( t) (Θ) m(a 0 ) /N + τ (t) (Θ) m(a ) /N (2.) where A t denotes the set of points which divide geodesics starting in A 0 and ending in A with ratio t:(-t) and where Θ denotes the minimal/maximal length of such geodesics, that is, A t := {y : (x 0, x ) A 0 A : d(y, x 0 ) = t d(x 0, x ), d(y, x ) = ( t) d(x 0, x )} and Θ := In particular, if K 0 then { infx0 A 0,x A d(x 0, x ), if K 0 sup x0 A 0,x A d(x 0, x ), if K < 0. m(a t ) /N ( t) m(a 0 ) /N + t m(a ) /N. (2.2) Proof. Let us first assume that 0 < m(a 0 ) m(a ) <. Applying the curvature-dimension condition CD(K, N) to ν i := m(a i ) A i m for i = 0, yields ρ t (y) /N dm(y) τ ( t) (Θ) m(a 0 ) /N + τ (t) (Θ) m(a ) /N (2.3) A t where ρ t denotes the density of some geodesic Γ t connecting ν 0 and ν. (Here without restriction we assume N ). Now by Jensen s inequality the LHS of (2.3) is dominated by m(a t ) /N. This proves the claim, provided m(a 0 ) m(a ) <. The general case follows by approximation of A i by sets of finite volume. Remark 2.2. The assumption m(a 0 ) m(a ) > 0 can not be dropped. For instance, let = R 2 and let m be the -dimensional Lebesgue measure on A 0 := {0} R and choose A = {} R. Now let us fix a point x 0 supp[m] and study the growth of the volume of concentric balls v(r) := m(b r (x 0 )) as well as the growth of the volume of the corresponding spheres s(r) := lim sup δ 0 δ m ( B r+δ (x 0 ) \ B r (x 0 ) ). Theorem 2.3 ( Generalized Bishop-Gromov Volume Growth Inequality ). Assume that the metric measure space (, d, m) satisfies the curvature-dimension condition CD(K, N) for some real numbers K, N R, N. Then each bounded set has finite volume. oreover, either m is supported by one point or all points and all spheres have mass 0.
N ore precisely, if N > then for each fixed x 0 supp[m] and all 0 < r < R K 0 π s(r) s(r) ( ) K sin N r ( ) K sin N R N (2.4) and v(r) v(r) r ( ) N 0 sin K N t dt ( ) N (2.5) 0 sin K N t dt R with s(.) and v(.) defined as above and with the usual interpretation of the RHS if K 0. In particular, if K = 0 s(r) ( r ) N s(r) v(r) ( r ) N and R v(r). R The latter also holds true if N = and K 0. For each K and each integer N > the simply connected spaces of dimension N and constant curvature K/(N ) provide examples where these volume growth estimates are sharp. But also for arbitrary real numbers N > these estimates are sharp as demonstrated by Example.8(i) and (ii) where equality is attained. Proof. Let us fix a point x 0 supp[m] and assume first that m({x 0 }) = 0. Let numbers r, R with 0 < r < R be given and put t = r R. Choose numbers ɛ > 0 and δ > 0. We will apply the generalized Brunn-inkowski inequality from above to A 0 := B ɛ (x 0 ) and A := B R+δR (x 0 ) \ B R (x 0 ). One easily verifies that A t B r+δr+ɛr/r (x 0 ) \ B r ɛr/r (x 0 ) and R ɛ Θ R + δr + ɛ. Hence, Proposition 2. implies m ( B r+δr+ɛr/r (x 0 ) \ B r ɛr/r (x 0 ) ) /N τ ( r/r) (R δr ɛ) m (B ɛ (x 0 )) /N +τ (r/r) (R δr ɛ) m ( B R+δR (x 0 ) \ B R (x 0 ) ) /N where has to be chosen to coincide with the sign of K. In the limit ɛ 0 this yields m ( B (+δ)r (x 0 ) \ B r (x 0 ) ) /N τ (r/r) (( δ)r) m ( B (+δ)r (x 0 ) \ B R (x 0 ) ) /N or, in other words, v(( + δ)r) v(r) τ (r/r) (( δ)r)n [v(( + δ)r) v(r)]. (2.6) For r small enough, the LHS will be finite since by assumption m is locally finite and thus v(r) will be finite for all R R + and it will coincide with v(r ) for all R R N := K 0 π. oreover, by construction v will be right continuous and nondecreasing with at most countably many discontinuities. In particular, there will be arbitrarily small r > 0 and δ > 0 such that v is continuous on the interval [r, ( + δ)r). Hence, by (2.6) v will be continuous on R +. Therefore, m( B r (x 0 )) = 0 for all r > 0 and in turn m({x}) = 0 for all x x 0. 2
Inequality (2.6) can be restated as [v(( + δ)r) v(r)] [v(( + δ)r) v(r)] sin( δr δr sin( K N K N ( δ)r) ( δ)r) with the usual interpretation if K 0. In the limit δ 0 this yields the first claim (2.4). Furthermore, given r and δ by successive subdivision of the interval [r, (+δ)r] one can construct a sequence (r n ) n of points in [r, ( + δ)r] with 0 [ v(( + 2 n 2 n δ)r) v(r) ] [v(( + δ)r) v(r)] =: C. δr δr Together with (2.7) this implies that v is locally Lipschitz continuous on R +. Therefore, in particular, it is weakly differentiable a.e. on R + and it coincides with the integral of its weak derivative s. We thus may apply Lemma 3. from [Ch93] according to which the inequality (2.4) implies the integrated version (2.5). It only remains to treat the case m({x 0 }) > 0. If there were a point x supp[m] \{x 0 } then we could apply the previous arguments (now with x in the place of x 0 ) and deduce that m({x}) = 0 for all x x which would lead to the contradiction m({x 0 }) = 0. Hence, m({x 0 }) > 0 implies supp[m] = {x 0 }. All the estimates of the theorem are trivially true in this case. Corollary 2.4 ( Doubling ). For each metric measure space (, d, m) which satisfies the curvature-dimension condition CD(K, N) for some real numbers K, N R, N, the doubling property holds on each bounded subset supp[m]. In particular, each bounded closed subset supp[m] is compact. If K 0 or N = the doubling constant is 2 N. Otherwise, it can be estimated in terms of K, N and the diameter L of as follows ( ) N K C 2 N cosh N L. Proof. Assume N > and put κ = property m(b 2r (x)) m(b r (x)) 2 N (2.7) ( K) 0 N. Then (2.5) immediately yields the doubling r 0 sinh (κ2t)n dt r 0 sinh (κt)n dt 2 N cosh(κr) N. The doubling property, however, always implies compactness of the support, see e.g. the proof of Theorem I.3.6. Corollary 2.5 ( Hausdorff Dimension ). For each metric measure space (, d, m) which satisfies the curvature-dimension condition CD(K, N) for some real numbers K, N R, N, the support of m has Hausdorff dimension N. Proof. We will prove that for each N > N the N -dimensional Hausdorff measure of vanishes. Without restriction we may assume that is bounded and that it has full support. For each ɛ > 0 we can estimate the ɛ-approximate N -dimensional Hausdorff measure of as follows: ( ) N HN ɛ () := c N inf 2 diams j : S j =, diams j ɛ j= j= k c N ɛ N inf k N : B ɛ (x j ) =. 3 j=
According to the doubling property as derived in Corollary 2.4, the minimal number k in the last term can be estimated by k C ɛ N (cf. also proof of Theorem I.4.9). Hence, for each N > N. H N () := lim ɛ 0 H ɛ N () = 0 Corollary 2.6 ( Generalized Bonnet-yers Theorem ). For every metric measure space (, d, m) which satisfies the curvature-dimension condition CD(K, N) for some real numbers K > 0 and N the support of m is compact and has diameter N L K π. In particular, if K > 0 and N = then supp[m] consists of one point. N Proof. Let two points x 0, x supp[m] and ɛ > 0 be given with d(x 0, x ) K π + 4ɛ and 0 < m(b ɛ (x i ) <. Put A i = B ɛ (x i ) for i = 0,. Then A /2 B R (x 0 ) for some finite R. Hence, Proposition 2. with Θ > N K π implies m(a /2) = whereas Theorem 2.3 implies N K π for all x 0, x supp[m]. m(b R (x 0 )) <. This contradiction shows that d(x 0, x ) Finite diameter, however, implies compactness of supp[m] according to Corollary 2.4. 3 Stability under Convergence Theorem 3.. Let (( n, d n, m n )) n N be a sequence of normalized metric measure spaces where each n N the space ( n, d n, m n ) satisfies the curvature-dimension condition CD(K n, N n ) and has diameter L n. Assume that for n ( n, d n, m n ) D (, d, m) and (K n, N n, L n ) (K, N, L) for some triple (K, N, L) R 2 satisfying K L 2 < (N )π 2. Then the space (, d, m) satisfies the curvature-dimension condition CD(K, N) and has diameter L. Corollary 3.2. For each triple (K, L, N) R 3 with K L 2 < (N )π 2 the family X (K, N, L) of isomorphism classes of normalized metric measure spaces which satisfy the curvature-dimension condition CD(K, N) and which have diameter L is compact w.r.t. D. Proof of the Corollary. Let numbers K, N, L as above be given. The volume growth estimate (2.5) implies that each element in X (K, N, L) satisfies the doubling property with some uniform doubling constant C = C(K, N, L). According to Theorem I.3.6, the family of all (, d, m) with doubling constant C and diameter L is compact. Hence, it suffices to prove that X (K, N, L) is closed under D-convergence. This is the content of the previous theorem. Given t [0, ], K R and N we introduce for the sequel the abbreviation T (t) (q m) = [ τ ( t) ( d(x 0, x )) ρ 0 (x 0 ) /N + τ (t) ( d(x 0, x )) ρ (x ) /N] dq(x 0, x ) whenever q is coupling of ν 0 = ρ 0 m and ν = ρ m. 4
Lemma 3.3. Let K, N R with N >. For each sequence q (k) of optimal couplings with the same marginals ν 0 and ν which converge to some coupling q ( ) lim sup T (t) (q(k) m) T (t) (q m). k Proof. Let q (k), k N, and q ( ) as above. We will prove that lim inf k τ ( t) ( d(x 0, x ))ρ 0 (x 0 ) /N dq (k) (x 0, x ) τ ( t) ( d(x 0, x ))ρ 0 (x 0 ) /N dq ( ) (x 0, x ). (3.) Together with an analogous assertion with ρ in the place of ρ 0 (and t in the place of t) this will prove the claim. For k N { } and C R + { } put v (k) C (x 0) = [ ] τ ( t) ( d(x 0, x )) C Q (k) (x 0, dx ) where Q (k) (x 0, dx ) denotes the disintegration of dq (k) (x 0, x ) w.r.t. dν(x 0 ). Now fix C R +. Since C b () is dense in L (, ν 0 ) and since 0 v (k) C (.) C, for each ɛ > 0 there exists a function ψ C b () such that [ ] v (k) ρ /N 0 C ψ dν 0 ɛ (3.2) C for all k N { }. The weak convergence q (k) q on implies that there exists a k(ɛ) N such that for each k k(ɛ): v ( ) C ψ dν 0 v (k) C ψ dν 0 + ɛ. (3.3) Summing up (3.2) and (3.3) we obtain [ ] v ( ) C ρ /N 0 C dν 0 v ( ) C v (k) C ψ dν 0 + ɛ v (k) C [ ] ρ /N 0 C dν 0 + 3ɛ ψ dν 0 + 2ɛ v (k) ρ /N 0 dν 0 + 3ɛ. That is, for each C R + [ ] v ( ) C ρ /N 0 C dν 0 lim inf k v (k) ρ /N 0 dν 0. Finally, as C monotone convergence yields v ( ) ρ /N 0 dν 0 lim inf k v (k) ρ /N 0 dν 0. This is precisely our claim (3.). Proof of the Theorem. (i) Let (( n, d n, m n )) n N be a sequence of normalized metric measure spaces as above, each of them satisfying a curvature-dimension condition CD(K n, N n ) and having diameter L n. oreover, assume that ( n, d n, m n ) (, d, m) as n. Then obviously also (, d, m) has diameter L. Without restriction, we may assume that N n > and that 5
there exists a triple (K 0, N 0, L 0 ) with K 0 L 2 0 < (N 0 )π 2 and K n K 0, L n L 0, N n N 0 for all n N. In order to verify the curvature-dimension condition CD(K, N) let two arbitrary measures ν 0 = ρ 0 m and ν = ρ m in P 2 (, d, m) and a number ɛ > 0 be given. (ii) Fix an arbitrary optimal coupling q of them and put E r := {(x 0, x ) 2 : ρ 0 (x 0 ) < r, ρ (x ) < r}, α r := q(e r ) and q (r) (.) := α r q(. E r ) for r R +. The latter has marginals ν (r) 0 (.) := q(r) (. ), ν (r) (.) := q(r) (.) with bounded densities. oreover, for sufficiently large r = r(ɛ) d W (ν 0, ν (r) 0 ) ɛ, d W (ν, ν (r) ) ɛ. (3.4) (iii) Since the densities of ν (r) 0 and ν (r) are bounded there exist a number R R such that sup Ent( ν (r) i m) + sup n K n d 2 W ( ν (r) i=0, 8 0, ν(r) ) R. (3.5) Choose n = n(ɛ) N and a coupling ˆd of the metrics d and d n with { ( 2 ˆd W (m n, m) D(( n, d n, m n ), (, d, m)) min exp 2 + 4L2 0 R ) } ɛ ɛ 2, 4C (3.6) for some constant C to be specified later. Following the proofs of Lemma I.4.9 and Theorem I.4.20, fix a coupling p of m and m n which is optimal w.r.t. ˆd and let P and P be disintegrations of p w.r.t. m and m n, resp. Recall that P defines a canonical map P : P 2 (, d, m) P 2 ( n, d n, m n ). Put ν i,n := P ( ν (r) i ) = ρ i,n m n with ρ i,n (y) = ρ (r) i (x)p (y, dx) for i = 0,. Then (3.5) and (3.6) imply, according to Lemma I.4.9, ˆdW ( ν (r) 0, ν 0,n) ɛ, ˆdW ( ν (r), ν,n) ɛ. (3.7) (iv) Due to the curvature-dimension condition on ( n, d n, m n ) there exist an optimal coupling q n of ν 0,n, ν,n and a geodesic Γ t,n connecting them and satisfying for all N N n, K K n and t [0, ]. Put S N (Γ t,n m n ) T (t) K,N (q n m n ) (3.8) Γ ɛ t = P (Γ t,n ) with n = n(ɛ) as above and P : P 2 ( n, d n, m n ) P 2 (, d, m) as introduced in Lemma I.4.9. Then essentially with the same arguments as in the proof of Lemma I.4.9 (now Jensen s inequality applied to the convex function r r /N ) S N (Γ ɛ t m) S N (Γ t,n m n ) (3.9) for all N and t under consideration. oreover, we know that the curvature-dimension condition CD(K n, N n ) implies the curvature bound Curv( n, d n, m n ) K n (cf. Proposition.6(ii)) which in turn implies Ent(Γ ɛ t m) Ent(Γ t,n m n ) R. 6
This (together with (3.6)) allows to apply again Lemma I.4.9 to deduce finally ˆdW (Γ ɛ t, Γ t,n ) ɛ. (3.0) (v) For fixed N, K and t put v 0 (y 0 ) = τ ( t) K,N ( d n (y 0, y )) Q n (y 0, dy ) n and v (y ) = τ (t) K,N ( d n (y 0, y )) Q n(y, dy 0 ) n where Q n and Q n are disintegrations of q n w.r.t. ν 0,n and ν,n, resp. Then oreover, T (t) K,N (q n m n ) = n v 0 (y 0 ) P (x 0, dy 0 ) = = = n n n ρ i,n (y) /N v i (y) dm n (y) n [ /N ρ (r) i (x) P (y, dx)] v i (y) dm n (y) i=0 i=0 i=0 i=0 n n n τ ( t) n n where C denotes the maximum of Analogously, v (y ) P (x, dy ) n n n ρ (r) i (x) /N P (y, dx) v i (y) dm n (y) ] ρ (r) i (x) [ /N v i (y) P (x, dy) dm(x). n K,N ( d n ((y 0, y ))Q n (y 0, dy ) P (x 0, dy 0 ) [ ] τ ( t) K,N ( d(x 0, x )) C ( d n (y 0, y ) d(x 0, x )) θ τ (s) ρ (r) (x ) ρ,n (y ) P (y, dx ) Q n (y 0, dy ) P (x 0, dy 0 ) [ τ ( t) K,N ( d(x 0, x )) C (ˆd(x0, y 0 ) + ˆd(x )], y ) ρ (r) (x ) ρ,n (y ) P (y, dx ) Q n (y 0, dy ) P (x 0, dy 0 ) K,N (θ) for s [0, ], N N 0, K K 0 and θ L 0. [ τ (t) K,N ( d(x 0, x )) C (ˆd(x0, y 0 ) + ˆd(x )], y ) ρ (r) 0 (x 0) ρ 0,n (y 0 ) P (y 0, dx 0 ) Q n(y, dy 0 ) P (x, dy ). Define a coupling q r (not necessarily optimal) of ν (r) 0 and ν (r) by (r) dq r ρ 0 (x 0, x ) = (x 0) ρ (r) (x ) ρ 0,n (y 0 )ρ,n (y ) P (y, dx ) P (y 0, dx 0 ) dq n (y 0, y ) = n n n n (r) ρ 0 (x 0) ρ (r) (x ) P (y, dx ) Q n (y 0, dy ) P (x 0, dy 0 ) m(dx 0 ). ρ,n (y ) 7
and a coupling q ɛ of ν 0 and ν by q ɛ (.) := α r q r + q(. ( 2 \ E r )) for r = r(ɛ). Then the above estimates yield [ T (t) K,N (q n m n ) T (t) K,N (q r m) + C ρ (r) 0 (x) /N + ρ (r) ] (x) /N ˆd(x, y) dp(x, y) T (t) K,N (q r m) + 2C ˆd W (m, m n ) T (t) K,N (q r m) + ɛ due to our choice of n. oreover, lim T (t) K,N (q ɛ m) T (t) K,N (q r(ɛ) m) = 0. (3.) ɛ 0 (vi) Summarizing, we have for each ɛ > 0 a probability measure q ɛ on 2 and a family of probability measures Γ ɛ t, t [0, ] on satisfying S N (Γ ɛ t m) S N (Γ t,n m n ) T (t) K,N (q n m n ) T (t) K,N (q r(ɛ) m) + ɛ. (3.2) Compactness of implies that there exists a sequence (ɛ(k)) k N converging to 0 such that the measures q ɛ(k) converge to some q and for each rational t [0, ] the measures Γ ɛ(k) t converge to some Γ t. The measure q has marginals ν 0 and ν. According to (3.0), (3.7) and (3.4), it is even an optimal coupling of them. For each n N the family Γ t,n, t [0, ] is a geodesic in P 2 ( n, d n, m n ) connecting ν 0,n and ν,n. As n the latter converge to ν 0 and ν, resp. Together with (3.0) this implies d W (Γ s, Γ t ) s t d W (ν 0, ν ) for all rational s, t [0, ]. Hence, the family (Γ t ) t extends to a geodesic connecting ν 0 and ν. oreover, (3.2) and (3.) together with lower semicontinuity of S N (. m) (Lemma.) and upper semicontinuity of T (t) K,N (. m) (Lemma 3.3) imply S N (Γ t m) lim inf k S N (Γɛ(k) t m) lim inf T (t) k K,N (q ɛ(k) m) T (t) K,N (q m) (3.3) for all t [0, ], all N > N = lim n N n and all K < K = lim n K n. By continuity of S N and T (t) K,N in (K, N ) the inequality S N (Γ t m) T (t) K,N (q m) also holds for (K, N ) = (K, N). This proves the Theorem. 4 Nonbranching Spaces Several aspects of optimal mass transportation become much simpler if the underlying space is nonbranching in the sense of Definition I.2.8. In this chapter, we will study the curvaturedimension condition for nonbranching spaces. Lemma 4.. Assume that (, d, m) is nonbranching and satisfies condition CD(K, N) for some pair (K, N). Then for every x supp[m] and m-a.e. y (with exceptional set depending on x) there exists a unique geodesic between x and y. oreover, there exists a measurable map γ : 2 G() such that for m m-a.e. (x, y) 2 the curve t γ t (x, y) is the unique geodesic connecting x and y. 8
Proof. Fix x 0, t ]0, [ and some closed set A. Let A n t for n N denote the set of all t-intermediate points z = γ t (x, y) between points x B /n (x 0 ) and y A. Assume without restriction m(b /n (x 0 )) m(a ) > 0. According to Corollary 2. m(a n t ) for each n and thus (as n ) with A t := n An t. inf ς (t) ( d(x, y)) m(a ) x B /n (x 0 ),y A m(a t ) inf y A ς (t) ( d(x, y)) m(a ) Each point z A t lies on some geodesic starting in x 0 and ending somewhere in A. Indeed, for each n the point z will be a t-intermediate point of some x n B /n (x 0 ) and some y n A. By local compactness of and closedness of A there exists a point y 0 A such that (after passing to a suitable subsequence) y n y 0 and thus z will also be a t-intermediate point of x 0 and y 0. Now choose A = B R (x 0 ) for some large R. Decomposing A into a disjoint union i Ai with A i = A ( B ɛi (x 0 ) \ B ɛ(i ) (x 0 ) ) and applying the previous estimate to each of the A i yields (as ɛ 0) m(a t ) A ς (t) ( d(x 0, y)) m(dy) where A t denotes the set of t-intermediate points between x 0 and some y A. Nonbranching of therefore will imply that for each z A t the geodesic from x 0 to z is unique. Now ς (t) ( d(x 0, y)) as t for all y with K d 2 (x 0, y) < (N )π 2 and ς (t) ( d(x 0, y)) = and for all other y. Hence, m(a \ t< A t) = 0 and thus for m- a.e. z A there exists a unique geodesic connecting x 0 and z. Finally, for R this yields the claim concerning uniqueness of geodesics. For the claim concerning the measurable choice of geodesics (or intermediate points), fix a number t ]0, [ and assume for simplicity m() =. For each k N let = i i,k be a (finite or countable) covering of by measurable sets i,k with diameter /k and λ i,k := m( i,k ) > 0. Let p i,k be a probability measure on 3 with the following properties: the projection (π ) p i,k onto the first component is the probability measure λ i,k i,k m; the projection on the third component is m; the joint distribution of the first and third component is an optimal coupling of them; and conditioned under the first and third component, the second component is a t-intermediate point of them. Hence, for each k the probability measure p k := i λ i,kp i,k on 3 has the following properties: the projection on the first component is m; the projection on the third component is m; and conditioned under the first and third component, the second component is a t-intermediate point of them. Now by compactness there exists an accumulation point p of the p k, k N. It has the following properties: the joint projection on the first and third component is m m; and conditioned under the first and third component, the second component is a t-intermediate point of them. That is, p(a C) = m(a) m(c) for all measurable A, C ; moreover, for p-a.e. (x, z, y) 3 the point z is a t-intermediate point of x and y. Disintegration of measures yields a arkov kernel P from 2 to such that dp(x, z, y) = P (x, y; dz)m(dx)m(dy). 9
According to the uniqueness of t-intermediate points P (x, y; dz) = δ γt (x,y)(dz) for m 2 -a.e. (x, y). This finally proves the measurability of γ t since by definition P is measurable in (x, y). Proposition 4.2. Given numbers K R and N and a compact nonbranching metric measure space (, d, m). Then the following are equivalent: (i) (, d, m) satisfies the curvature-dimension condition CD(K, N); (ii) For each pair ν 0, ν P 2 (, d, m) there exist a geodesic Γ : [0, ] P 2 (, d, m) connecting ν 0, ν and an optimal coupling q such that for all t [0, ] and all N N S N (Γ(t) m) τ ( t) (Θ) S N (ν 0 m) + τ (t) (Θ) S N (ν m) (4.) { } { } q-essinfx0,x where Θ := d(x 0, x ), if K 0 minimal denotes the transportation q-esssup x0,x d(x 0, x ), if K < 0 maximal distance. (iii) For each pair of points z 0, z there exists an ɛ > 0 such that for each pair ν 0, ν P 2 (, d, m) with supp[ν 0 ] B ɛ (z 0 ), supp[ν ] B ɛ (z ) there exist an optimal coupling q and a geodesic Γ : [0, ] P 2 (, d, m) connecting them and satisfying (.). (iv) For each pair ν 0, ν P 2 (, d, m) and each optimal coupling q of them ρ t (γ t (x 0, x )) [ τ ( t) ( d(x 0, x )) ρ /N 0 (x 0 ) + τ (t) ( d(x 0, x )) ρ /N (x )] N (4.2) for all t [0, ], and q-a.e. (x 0, x ) 2. Here ρ t is the density of the push forward of q under the map (x 0, x ) γ t (x 0, x ). It is determined by u(y)ρ t(y) dm(y) = u(γ t(x 0, x )) dq(x 0, x ) for all bounded measurable u : R. (v) For each pair ν 0, ν P 2 (, d, m) and each optimal coupling q of them there exists a geodesic Γ : [0, ] P 2 (, d, m) connecting ν 0, ν and satisfying (.) for all t [0, ] and all N N. Proof. (i) (iii), (iv) (i), and (v) (i): trivial. (i) (ii): Immediate consequence of the fact that τ (t) ( d(x 0, x )) τ (t) (Θ) for all t [0, ] and all x 0, x with K d(x 0, x ) KΘ. (Actually, this implication does not require that is compact and nonbranching.) (iii) (i): By compactness of, there exist finitely many disjoint sets L,..., L n which cover such that for each pair i, j {,..., n} and each pair ν 0, ν P 2 (, d, m) with supp[ν 0 ] L i, supp[ν ] L j there exist an optimal coupling q and a geodesic Γ : [0, ] P 2 (, d, m) connecting them and satisfying (.). Now let arbitrary ν 0 and ν P 2 (, d, m) be given. Fix an arbitrary optimal coupling q of them and define probability measures ν ij 0 ν ij 0 (A) := and νij q((a L i ) L j ) and ν ij α ij for i, j =,..., n by (A) := α ij q(l i (A L j )) (4.3) provided α ij := q(l i L j ) 0. Then supp[ν ij 0 ] L i and supp[ν ij ] L j. Therefore, for each pair (i, j) {,..., n} 2 the assumption can be applied to the probability measures ν ij 0 and νij. 20
It yields the existence of an optimal coupling q ij of them and of a geodesic Γ ij connecting them with the property [ S N (Γ ij t m) τ ( t) ( d(x 0, x )) ρ ij 0 (x 0) /N for all t [0, ] and all N N. Define q := + τ (t) ( d(x 0, x )) ρ ij (x ) /N ] dq ij (x 0, x ) (4.4) n α ij q ij, Γ t := i,j= n i,j= α ij Γ ij t. (4.5) Then q is an optimal coupling of ν 0 and ν and Γ is a geodesic connecting them. oreover, since the ν ij 0 νij for different choices of (i, j) {,..., n}2 are mutually singular and since is nonbranching, also the Γ ij t for different choices of (i, j) {,..., n} 2 are mutually singular, Lemma I.2.(iii) (for each fixed t [0, ]). Hence, S N (Γ t m) = i,j α /N ij S N (Γ ij t m) and one simply may sum up both sides of inequality (4.4) multiplied by α /N ij the claim. to obtain (ii) (i): Let numbers K, N and a compact nonbranching space (, d, m) with the property (4.) be given. oreover, let two measures ν 0 = ρ 0 and ν = ρ m P 2 (, d, m) be given and choose an arbitrary optimal coupling q of them. For each ɛ > 0 choose a finite covering (L i ) i=,...,n of by sets L i of diameter ɛ/2. Define numbers α ij and probability measures ν ij 0 and ν ij for i, j =,..., n as in the previous proof. Then by assumption there exist an optimal coupling q ij of them and a geodesic Γ ij connecting them with the property [ S N (Γ ij t m) τ ( t) ( d(x 0, x ) ɛ) ρ ij 0 (x 0) /N + τ (t) ( d(x 0, x ) ɛ) ρ ij (x ) /N ] dq ij (x 0, x ) (4.6) for all t [0, ] and all N N and with depending on the sign of K. Then for each ɛ > 0 as before q (ɛ) := n i,j= α ijq ij defines an optimal coupling of ν 0 and ν and Γ (ɛ) t := n i,j= α ijγ ij t defines a geodesic connecting ν 0 and ν. Compactness of implies that there exists a sequence (ɛ(k)) k N converging to 0 such that q (ɛ(k)) converge to some q and such that the geodesics Γ (ɛ(k)). converge to some geodesic Γ. in P 2 (, d, m). Hence, for each fixed ɛ > 0 and all t and N > under consideration S N (Γ t m) lim inf k lim sup k S N (Γ(ɛ(k)) t m) [ τ ( t) ( d(x 0, x ) ɛ ) ρ (ɛ(k)) 0 (x 0 ) /N + τ (t) ( d(x 0, x ) ɛ ) ρ (ɛ(k)) (x ) /N ] dq (ɛ(k)) (x 0, x ) [ τ ( t) ( d(x 0, x ) ɛ ) ρ 0 (x 0 ) /N + τ (t) ( d(x 0, x ) ɛ ) ρ (x ) /N ] dq(x 0, x ) 2
where the last inequality is proven similarly as Lemma 3.3. Finally, by monotone convergence the claim follows as ɛ 0. (i) (iv): Assume that the CD(K, N) condition holds and that is compact and nonbranching. Let measures ν 0 and ν be given as well as an optimal coupling q of them. Choose a -stable generator { n } n N of the Borel σ-field of with m( n ) = 0 for all n. For each n N consider the disjoint covering of by the 2 n sets L =... n, L 2 =... n n,..., L 2 n =... n. For each fixed n, define probability measures ν ij 0 and νij as in (4.3) (proof of the implication (iii) (i) ) and choose optimal couplings q ij of them with [ τ ( t) ( d(x 0, x )) ρ ij 0 (x 0) /N + τ (t) ( d(x 0, x )) ρ ij (x ) /N ] dq ij (x 0, x ) Define as in (4.5) ρ ij t (γ t(x 0, x )) /N dq ij (x 0, x ). (4.7) q (n) := 2 n i,j= α ij q ij. (4.8) Then by construction for all i, j n [ τ ( t) ( d(x 0, x )) ρ (n) 0 (x 0) /N + τ (t) ( d(x 0, x )) ρ (n) (x ) /N ] dq (n) (x 0, x ) i j ρ (n) t (γ t (x 0, x )) /N dq (n) (x 0, x ). (4.9) i j Compactness of implies that at least along a suitable subsequence the q (n) converge to an optimal coupling q of ν 0 and ν. Since m( i ) = 0 for all i, we obtain for all i, j N q( i j ) = lim n q(n) ( i j ) = q( i j ). Hence, q = q. oreover, we may apply (modifications of) Lemma. and Lemma 3.3 to pass to the limit in (4.9) and to obtain [ τ ( t) ( d(x 0, x )) ρ 0 (x 0 ) /N + τ (t) ( d(x 0, x )) ρ (x ) /N ] dq(x 0, x ) i j ρ t (γ t (x 0, x )) /N dq(x 0, x ). i j Since this holds for all i, j it finally implies τ ( t) ( d(x 0, x )) ρ 0 (x 0 ) /N + τ (t) ( d(x 0, x )) ρ (x ) /N ρ t (γ t (x 0, x )) /N for q-a.e. (x 0, x ) 2. With the particular choice N = N this is (iv). (iv) (v): We will prove that estimate (4.2) for a given N implies the corresponding estimate for any N N. Indeed, by Hölder s inequality and Lemma.2 [ ] ρ /N t (γ t (x 0, x )) τ ( t) ( d(x 0, x )) ρ /N 0 (x 0 ) + τ (t) ( d(x 0, x )) ρ /N N/N (x ) τ ( t) ( d(x 0, x )) N/N ( t) N/N ρ /N 0 (x 0 ) +τ (t) ( d(x 0, x )) N/N t N/N ρ /N (x ) τ ( t) ( d(x 0, x )) ρ /N 0 (x 0 ) + τ (t) ( d(x 0, x )) ρ /N (x ). 22