Münster J. of Math. 4 (2011), 53 64 Münster Journal of Mathematics urn:nbn:de:hbz:6-32449569769 c Münster J. of Math. 2011 lexandrov meets Lott-Villani-Sturm nton Petrunin (Communicated by Burkhard Wilking) bstract. Here I show the compatibility of two definitions of generalized curvature bounds: the lower bound for sectional curvature in the sense of lexandrov and the lower bound for Ricci curvature in the sense of Lott-Villani-Sturm. Introduction Let me denote by CD[m,κ] the class of metric-measure spaces which satisfy a weak curvature-dimension condition for dimension m and curvature κ (see preliminaries). By lex m [κ], I will denote the class of all m-dimensional lexandrov spaces with curvature κ equipped with the volume-measure (so lex m [κ] is a class of metric-measure spaces). Main theorem. lex m [0] CD[m,0]. The question was first asked by Lott and Villani in [6, Rem. 7.48]. In [12], Villani formulates it more generally: lex m [κ] CD[m,(m 1) κ]. The latter statement can be proved along the same lines, but I do not write it down. bout the proof. The idea of the proof is the same as in the Riemannian case (see [4, Thm. 6.2] or [6, Thm. 7.3]). One only needs to extend certain calculus to lexandrov spaces. To do this, I use the same technique as in [10]. I will illustrate the idea on a very simple problem. Let M be a 2-dimensional nonnegatively curved Riemannian manifold and γ τ : [0,1] M be a continuous family of unit-speed geodesics such that (1) γ τ0 (t 0 )γ τ1 (t 1 ) t 1 t 0. Set l(t) to be the total length of the curve σ t : τ γ τ (t). Then l(t) is a concave function; that is easy to prove.
54 nton Petrunin Now, assume you have lex 2 [0] instead of M and a noncontinuous family of unit-speed geodesics γ τ (t) which satisfies (1). Define l(t) as the 1-dimensional Hausdorff measure of the image of σ t. Then l is also concave. Here is an idea how one can proceed; it is not the simplest one but the one which admits a proper generalization. Consider two functions ψ = dist Imσ0 and ϕ = dist Imσ1. Note that geodesics γ τ (t) are also gradient curves of ψ and ϕ. This implies that ϕ + ψ vanishes almost everywhere on the image of the map (τ,t) γ τ (t) (the Laplacians ϕ and ψ are Radon sign-measures). Then the result follows from my second variation formula from [9] and calculus on lexandrov spaces developed by Perelman in [8]. Remark. lthough CD[m, κ] is a very natural class of metric-measure spaces, some basic tools in Ricci comparison cannot work there in principle. For instance, there are CD[m, 0]-spaces which do not satisfy the bresch-gromoll inequality, (see[1]). Thus,onehastomodifythedefinitionoftheclassCD[m,κ] to make it suitable for substantial applications in Riemannian geometry. I am grateful to. Lytchak and C. Villani for their help. 1. Preliminaries Prerequisite. The reader is expected to be familiar with basic definitions and notions of optimal transport theory as in Villani s book [12], measure theory on lexandrov spaces from paper of Burago, Gromov and Perelman [3], DC-structure on lexandrov s spaces from Perelman s [8] and technique and notations of gradient flow as my survey [11]. What needs to be proved. Let me recall the definition of the class CD[m,0] only; it is sufficient for understanding this paper. The definition of CD[m, κ] can be found in [12, Def. 29.8]. Similar definitions were given by Lott and Villani in [6] and by Sturm in [13]. The idea behind these definitions, convexity of certain functionals in the Wasserstein space over a Riemannian manifold, appears in [7] by Otto an Vilani, [4] by Cordero-Erausquin, McCann and M. Schmuckenschläger, [14] by von Renesse and Sturm. In the Euclidean context, this notion of convexity goes back to [5] by McCann. More on the history of the subject can be found in the Villani s book [12]. For a metric-measure space X, I will denote by xy the distance between points x,y X and by vole the distinguished measure of Borel subset E X (I will call it volume). Let us denote by P 2 X the set of all probability measures with compact support in X equipped with Wasserstein distance of order 2, see [12, Def. 6.1]. Further, we assume X is a proper geodesic space; in this case P 2 X is geodesic. Let µ be a probability measure on X. Denote by µ r the absolutely continuous part of µ with respect to volume. I.e. µ r coincides with µ outside a Borel subset of volume zero and there is a Borel function : X R such that
lexandrov meets Lott-Villani-Sturm 55 µ r = vol. Define U m µ == def X 1 1 m dvol = X 1 dµr. m Then X CD[m,0] if the functional U m is concave on P 2 X; i.e. for any two measures µ 0,µ 1 P 2 X, there is a geodesic path µ t, in P 2 X, t [0,1], such that the real function t U m µ t is concave. Calculus in lexandrov spaces. Let lex m [κ]ands bethesubset of singular points; i.e. x S if and only if its tangent space T x is not isometric to Euclidean m-space E m. The set S has zero volume ([3, Thm. 10.6]). The set of regular points \S is convex ([9]); i.e. any geodesic connecting two regular points consists only of regular points. ccording to [8], if f : R is a semiconcave function and Ω is an image of a DC 0 -chart, then k f and the components of the metric tensor g ij are functions of locally bounded variation which are continuous in Ω\ S. Further, for almost all x the Hessian of f is well-defined. I.e. there is a subset of full measure Regf \S such that for any p Regf there is a bilinear form 1 Hess p f on T p such that f(q) = f(p)+d p f(v)+hess p f(v,v)+o( v 2 ), where v = log p q. Moreover, the Hessian can be found using standard calculus in the DC 0 -chart. In particular, TraceHessf == a.e. i(detg g ij j f). detg The following is an extract from the second variation formula[9, Thm. 1.1B] reformulated with formalism of ultrafilters. Let Ñ be a nonprincipal ultrafilter on the natural numbers, lex m [0] and [pq] be a minimizing geodesic in which is extendable beyond p and q. ssume further that one of (and therefore each) of the points p and q is regular. Then there is a model configuration p, q E m and isometries ı p : T p T p E m, ı q : T q T q E m such that for any fixed v T p and w T q we have ( exp n v) 1 ( p exp n w) 1 ( q exp p ı 1 p n v) ( exp q ı 1 q n w) +o(n 2 ) for Ñ-almost all n (once the left-hand side is well-defined). If τ : T p T q is the parallel translation in E m, then the isometry τ : T p T q which satisfies the identity ı q τ = τ ı p will be called the parallel transportation from p to q. Laplacians of semiconcave functions. Here are some facts from my paper on harmonic functions [10]. 1 Note that p \S, thus Tp is isometric to Euclidean m-space.
56 nton Petrunin Given a function f : R, define its Laplacian f to be a Radon signmeasure which satisfies the following identity u d f = u, f dvol for any Lipschitz function u : R. 1.1. Claim. Let lex m [κ] and f : R be λ-concave Lipschitz function. Then the Laplacian f is well-defined and f m λ vol. In particular, s f the singular part of f is negative. Moreover, f = TraceHessf vol+ s f. Proof. Let us denote by F t : the f-gradient flow for the time t. Given a Lipschitz function u : R, consider the family u t (x) = u F t (x). Clearly, u 0 u and u t is Lipschitz for any t 0. Further, for any x we have d + dt u t(x) t=0 Const. Moreover Further, d + dt u t(x) t=0 a.e. == d x u( x f) a.e. == x u, x f. u t dvol = u d(f t #vol), where # stands for push-forward. Since F t (x)f t (y) e λt xy (see [11, 2.1.4(i)]) for any x,y we have F t #vol exp( m λ t) vol. Therefore, for any nonnegative Lipschitz function u : R, u t dvol = u d(f t #vol) exp( m λ t) u dvol. Therefore u, f dvol = d+ dt u t dvol m λ u dvol. t=0 I.e. there is a Radon measure χ on such that u dχ = [ u, f +m λ u] dvol Set f = χ+m λ, this is a Radon sign-measure and χ = f +m λ 0.
lexandrov meets Lott-Villani-Sturm 57 Toprovethesecondpartofthetheorem, assumeuisanonnegativelipschitz functionwithsupportinadc 0 -chartu, whereu R m isanopensubset. Then Thus u, f = U detg g ij m i u j f dx 1 dx 2 dx = u i (detg g ij j f) dx 1 dx 2 dx m. U f = i (detg g ij j f) dx 1 dx 2 dx m a.e. == TraceHessf. Gradient curves. Here I extend the notion of gradient curves to families of functions, see [11] for all necessary definitions. Let I be an open real interval and λ : I R be a continuous function. one parameter family of functions f t : R, t I, will be called λ(t)-concave if the function (t,x) f t (x) is locally Lipschitz and f t is λ(t)-concave for each t I. We will write α ± (t) = f t if for any t I, the right/left tangent vector α ± (t) is well-defined and α ± (t) = α(t) f t. The solutions of α + (t) = f t will be also called f t -gradient curves. The following is a slight generalization of [11, 2.1.2 and 2.2(2)]; it can be proved along the same lines. 1.2. Proposition-Definition. Let lex m [κ], let I be an open real interval, let λ : I R be a continuous function and let f t : R, t I be a λ(t)-concave family. Then for any x and t 0 I there exists an f t -gradient curve α which is defined in a neighborhood of t 0 and such that α(t 0 ) = x. Moreover, if α,β : I are f t -gradient then for any t 0,t 1 I, t 0 t 1, α(t 1 )β(t 1 ) L α(t 0 )β(t 0 ), ( ) t1 where L = exp t 0 λ(t) dt. Note that the above proposition implies that the value α(t 0 ) of an f t - gradient curve α(t) uniquely determines it for all t t 0 in I. Thus we can define the f t -gradient flow as a family of maps F t0,t 1 : such that F t0,t 1 (α(t 0 )) = α(t 1 ) if α + (t) = f t. 1.3. Claim. Let f t : R be a λ(t)-concave family and F t0,t 1 be f t -gradient flow. Let E be a bounded Borel set. Fix t 1 and consider the function
58 nton Petrunin v(t) = volf 1 t,t 1 (E). Then v t1 t = t 1 t f ţ [ F 1 ţ,t 1 (E) ] dţ Proof. Let u : R be a Lipschitz function with compact support. Set u t = u F t,t1. Clearly every (x,t) u t (x) is locally Lipschitz. Thus, the function w u : t u t dvol is locally Lipschitz. Further w u(t) == a.e. u t, f t dvol = u t d f t. Therefore w u t 1 t = t 1 t dţ u ţ d f ţ. The last formula extends to an arbitrary Borel function u : R with bounded support. pplying it to the characteristic function of E we get the result. 2. Games with Hamilton-Jacobi shifts. Let lex m [0]. For a function f : R {+ }, let us define its Hamilton-Jacobi shift 2 H t f : R for the time t > 0 as follows { (H t f)(x) == def inf f(y)+ 2t xy 2} 1. y We say that H t f is well-defined if the above infimum is > everywhere in. Note that (2) H t0+t 1 f = H t1 H t0 f for any t 0,t 1 > 0. (The inequality H t0+t 1 f H t1 H t0 f is a direct consequence of the triangle inequality and it is actually equality for any intrinsic metric, in particular for lexandrov space.) Note that for t > 0, f t = H t f forms a 1 t-concave family, thus, we can apply Proposition-Definition 1.2 and Claim 1.3. The next theorem gives a more delicate property of the gradient flow for such families; it is an analog of [11, 3.3.6]. 2.1. Claim. Let lex m [0], f 0 : R be a function and let f t = H t f 0 be well-defined for t (0,1). ssume that γ : [0,1] is a geodesic path which is an f t -gradient curve for t (0,1) and that α : (0,1) is another 2 There is a lot of similarity between the Hamilton-Jacobi shift of a function and an equidistant for a hypersurface.
lexandrov meets Lott-Villani-Sturm 59 f t -gradient curve. Then if for some t 0 (0,1), α(t 0 ) = γ(t 0 ), then α(t) = γ(t) for all t (0,1). Proof. Set l = l(t) = α(t)γ(t) ; this is a locally Lipschitz function defined on (0, 1). We σ have to show that if l(t 0 ) = 0 for some t 0, then p l(t) = 0 for all t. α ccording to Proposition-Definition 1.2, l l(t) = 0 for all t t 0. In order to prove that x y z γ l(t) = 0 for all t t 0, it is sufficient to show that l [ 1 t + 2 1 t ] l for almost all t. Since α is locallylipschitz, foralmost all t, α + (t) and α (t) arewell-defined and opposite 3 to each other. Fix such t and set x = γ(0), z = γ(t), y = γ(1), p = α(t), so l(t) = pz. Note that the function (3) f t + 1 2(1 t) dist2 y has a minimum at z. Extend a geodesic [zp] by a both-sides infinite unitspeed quasigeodesic 4 σ : R, so σ(0) = z and σ + (0) = [zp]. The function f t σ : R R is 1 t-concave and from (3), It follows that f t σ(s) f t (z)+ γ + (t), [zp] s 1 2(1 t) s2. p f t,σ + (l) d p f t (σ + (l)) = (f t σ) + (l) γ + (t), [zp] [ 1 t + 2 1 t ] l. Now, (a) the vectors σ ± (l) are polar, thus α ± (t),σ + (l) + α ± (t),σ (l) 0, (b) the vectors α ± (t) are opposite, thus α + (t),σ ± (l) + α (t),σ ± (l) = 0, (c) α + (t) = p f t and σ (l) = [pz]. Thus, p f t,σ + (l) + α + (t), [pz] = 0. Therefore l = α + (t), [pz] γ + (t), [zp] [ 1 t + 2 1 t ] l. 2.2. Proposition. Let lex m [0], let f : R be a bounded continuous function and let f t = H t f. ssume that γ : (0,a) is an f t -gradient curve which is also a constant-speed geodesic. ssume that the function h(t) def == TraceHess γ(t) f t 3 I.e. α + (t) = α (t) and (α + (t),α (t)) = π 4 careful proof of existence of quasigeodesics can be found in [11].
60 nton Petrunin is defined for almost all t (0,a). Then h 1 m h2 in the sense of distributions; i.e. for any nonnegative Lipschitz function u : (0,a) R with compact support, a 0 ( 1 m h2 u h u ) dt 0. Proof. Since h is defined a.e., all T γ(t) for t (0,a) are isometric to Euclidean m-space. From (2), } f t1 (x) = inf {f t0 (y)+ xy 2. y 2 (t 1 t 0 ) Thus, for a parallel transportation τ : T γ(t0) T γ(t1) along γ, we have Hess γ(t1)f t1 (y) Hess γ(t0)f t0 (x)+ τ(x)y 2 2 (t 1 t 0 ) for any x T γ(t0) and y T γ(t1). Taking trace leads to the result. 3. Proof of the main theorem Let lex m [0]; in particular is a proper geodesic space. Let µ t be a family of probability measures on for t [0,1] which forms a geodesic path 5 in P 2 and both µ 0 and µ 1 are absolutely continuous with respect to volume on. It is sufficient 6 to show that the function Θ : t U m µ t is concave. ccording to [12, 7.22], there is a probability measure Π on the space of all geodesic paths in which satisfies the following: If Γ = suppπ and e t : Γ is evaluation map e t : γ γ(t) then µ t = e t #Π. The measure Π is called the dynamical optimal coupling for µ t and the measure π = (e 0,e 1 )#Π is the corresponding optimal transference plan. The space Γ will be considered further equipped with the metric γγ = max t [0,1] γ(t)γ (t). First we present µ t as the push-forward for gradient flows of two opposite families of functions. ccording to [12, 5.10], there are optimal price functions ϕ,ψ : R such that ϕ(y) ψ(x) 1 2 xy 2 5 i.e. a constant-speed minimizing geodesic defined on [0,1]. 6 It follows from [12, 30.32] since lexandrov s spaces are nonbranching.
lexandrov meets Lott-Villani-Sturm 61 for any x,y and equality holds for any (x,y) suppπ. We can assume that ψ(x) = + for x suppµ 0 and ϕ(y) = for y suppµ 1. Consider two families of functions ψ t = H t ψ and ϕ t = H 1 t ( ϕ). Clearly, ψ t forms a 1 t -concave family for t (0,1] and ϕ t forms 7 a 1 1 t -concave family for t [0,1). It is straightforward to check that for any γ Γ and t (0,1) in particular, ± γ ± (t),v = d γ(t) ψ t (v) = d γ(t) ϕ t (v); (4) γ + (t) = ψ t and γ (t) = ϕ t. For 0 < t 0 t 1 1, let us consider the maps Ψ t0,t 1 :, the gradient flow of ψ t, defined by Ψ t0,t 1 α(t 0 ) = α(t 1 ) if α + (t) = ψ t. Similarly, 0 t 0 t 1 < 1, define map Φ t1,t 0 : Φ t1,t 0 β(t 1 ) = β(t 0 ) if β (t) = ϕ t. ccording to Proposition-definition 1.2, (5) Ψ t0,t 1 is t1 t 0 -Lipschitz and Φ t1,t 0 is 1 t0 1 t 1 -Lipschitz. From (4), e t1 = Ψ t0,t 1 e t0 and e t0 = Φ t1,t 0 e t1. Thus, for any t (0,1), the map e t : Γ is bi-lipschitz. In particular, for any measure χ on, there is a uniquely determined one-parameter family of pull-back measures χ t on Γ, i.e. such that χ te = χ(e t E) for any Borel subset E Γ. Fix some z 0 (0,1) (one can take z 0 = 1 2 ) and equip Γ with the measure ν = vol z 0. Thus, from now on almost everywhere has sense in Γ, Γ (0,1) and so on. Now we will represent Θ in terms of families of functions on Γ. Note that µ 1 = Ψ t,1 #µ t and Ψ t,1 is 1 t -Lipschitz. Since µ 1 is absolutely continuous, so is µ t for all t. Set µ t = t vol. Note that from (5), we get that ( ) m ( ) m 1 t1 t 1 (γ(t 1 )) 1 t 0 t0 (γ(t 0 )) t1 t 0 for almost all γ Γ and 0 < t 0 < t 1 < 1. For γ Γ set r t (γ) = t(γ(t)). Then (6) Θ(t) = 1 m t dµ t = r 1 m t dπ. In particular, Θ is locally Lipschitz in (0,1). Γ 7 Note thatusuallyϕt isdefined withopposite sign, butiwanted toworkwith semiconcave functions only.
62 nton Petrunin Next we show that the measure ϕ t is absolutely continuous on e t Γ and that r t (γ(t)) = t(γ(t)) ϕ t in some weak sense. From (5), vol t = ewt ν for some Borel function w t : Γ R. Thus vole t E = e wt dν for any Borel subset E Γ. Moreover, for almost all γ Γ, we have that the function t w t (γ) is locally Lipschitz in (0,1) (more precisely, t w t (γ) coincides with a Lipschitz function outside of a set of zero measure). In particular w t t is well-defined a.e. in Γ (0,1) and moreover t a.e. w t == z 0 E w ţ ţ dţ. Further, from Claim 2.1, if 0 < t 0 t 1 < 1 then for any γ Γ, Thus, for any Borel subset E Γ, Set From Claim 1.3, Ψ t0,t 1 (x) = γ(t 1 ) x = γ(t 0 ), Φ t1,t 0 (x) = γ(t 0 ) x = γ(t 1 ). e t1 E = Ψ t0,t 1 e t0 E = Φ 1 t 1,t 0 (e t0 E), e t0 E = Φ t1,t 0 e t1 E = Ψ 1 t 0,t 1 (e t1 E). v(t) == def vole t E = E e wt dν. v (t) == a.e. ψ t (e t E) == a.e. ϕ t (e t E). Thus, ψ t + ϕ t = 0 everywhere on e t Γ. From Claim 1.1, ψ t m t vol, ϕ t m 1 t vol. Thus, both restrictions ψ t etγ and ϕ t etγ are absolutely continuous with respect to volume. Therefore v (t) == a.e. TraceHessϕ t dvol. e te For the one parameter family of functions h t (γ) = TraceHess γ(t) ϕ t, we have v t t z 0 = (e wt 1) dν = dţ h ţ e wţ dν z 0 or any Borel set E Γ. Equivalently, E w t t a.e. == h t E
lexandrov meets Lott-Villani-Sturm 63 From Proposition 2.2, h t t m h2 1 t. Thus, for almost all γ Γ, the following inequality holds in the sense of distributions: 2 ( ) ( t 2 exp wt (γ) 1 = m m 2 h 2 t + m h ) ( ) 1 t wt (γ) exp 0; t m ( ) ( ) i.e. t exp wt(γ) m is concave. Moreover precisely, t exp wt(γ) m coincides with a concave function almost everywhere. Clearly, for any t we have µ = r t e wt ν. Thus, for almost all γ there is a a.e. nonnegative Borel function a : Γ R 0 such that r t == a e wt. Continue (6), Θ(t) = r 1 m t dπ = e w t m m a dπ Γ I.e. Θ is concave as an average of concave functions. gain, more precisely, Θ coincides with a concave function a.e., but since Θ is locally Lipschitz in (0,1) we get that Θ is concave. Γ References [1] U. bresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. mer. Math. Soc. 3 (1990), no. 2, 355 374. MR1030656 (91a:53071) [2] J. Bertrand, Existence and uniqueness of optimal maps on lexandrov spaces, dv. Math. 219 (2008), no. 3, 838 851. MR2442054 (2009j:49094) [3] Yu.Burago, M.Gromov and G.Perel man, leksandrov spaces with curvatures bounded below. (Russian) Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3 51, 222; translation in Russian Math. Surveys 47 (1992), no. 2, 1 58. MR1185284 (93m:53035) [4] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), no. 2, 219 257. MR1865396 (2002k:58038) [5] R. J. McCann, convexity principle for interacting gases, dv. Math. 128 (1997), no. 1, 153 179. MR1451422 (98e:82003) [6] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, nn. of Math. (2) 169 (2009), no. 3, 903 991. MR2480619 (2010i:53068) [7] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. nal. 173 (2000), no. 2, 361 400. MR1760620 (2001k:58076) [8] G. Perelman, DC Structure on lexandrov Space, available on http://www.math.psu.edu/petrunin/papers/papers.html [9]. Petrunin, Parallel transportation for lexandrov space with curvature bounded below, Geom. Funct. nal. 8 (1998), no. 1, 123 148. MR1601854 (98j:53048) [10]. Petrunin, Harmonic functions on lexandrov spaces and their applications, Electron. Res. nnounc. mer. Math. Soc. 9 (2003), 135 141. MR2030174 (2005b:53067) [11]. Petrunin, Semiconcave functions in lexandrov s geometry, in Surveys in differential geometry. Vol. XI, 137 201, Surv. Differ. Geom., 11 Int. Press, Somerville, M. MR2408266 (2010a:53052) [12] C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer, Berlin, 2009. MR2459454 (2010f:49001)
64 nton Petrunin [13] K.-T. Sturm, On the geometry of metric measure spaces. I, cta Math. 196 (2006), no. 1, 65 131. MR2237206 (2007k:53051a) [14] M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure ppl. Math. 58 (2005), no. 7, 923 940. MR2142879 (2006j:53048) Received February 27, 2009; accepted July 19, 2010 nton Petrunin Department of Mathematics Penn State University University Park P 16802, US E-mail: petrunin@math.psu.edu URL: http://www.math.psu.edu/petrunin/