Alexandrov meets Lott-Villani-Sturm
|
|
- Peregrine Bennett
- 6 years ago
- Views:
Transcription
1 Münster J. of Math. 4 (2011), Münster Journal of Mathematics urn:nbn:de:hbz: c Münster J. of Math 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.
2 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
3 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.
4 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 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.
5 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, and 2.2(2)]; it can be proved along the same lines 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 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
6 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] 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.
7 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 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 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 t ] l 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].
8 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.
9 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.
10 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
11 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, MR (91a:53071) [2] J. Bertrand, Existence and uniqueness of optimal maps on lexandrov spaces, dv. Math. 219 (2008), no. 3, MR (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, MR (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, MR (2002k:58038) [5] R. J. McCann, convexity principle for interacting gases, dv. Math. 128 (1997), no. 1, MR (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, MR (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, MR (2001k:58076) [8] G. Perelman, DC Structure on lexandrov Space, available on [9]. Petrunin, Parallel transportation for lexandrov space with curvature bounded below, Geom. Funct. nal. 8 (1998), no. 1, MR (98j:53048) [10]. Petrunin, Harmonic functions on lexandrov spaces and their applications, Electron. Res. nnounc. mer. Math. Soc. 9 (2003), MR (2005b:53067) [11]. Petrunin, Semiconcave functions in lexandrov s geometry, in Surveys in differential geometry. Vol. XI, , Surv. Differ. Geom., 11 Int. Press, Somerville, M. MR (2010a:53052) [12] C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer, Berlin, MR (2010f:49001)
12 64 nton Petrunin [13] K.-T. Sturm, On the geometry of metric measure spaces. I, cta Math. 196 (2006), no. 1, MR (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, MR (2006j:53048) Received February 27, 2009; accepted July 19, 2010 nton Petrunin Department of Mathematics Penn State University University Park P 16802, US petrunin@math.psu.edu URL:
arxiv: v2 [math.mg] 10 Apr 2015
lexandrov meets Lott Villani Sturm nton Petrunin arxiv:1003.5948v2 [math.mg] 10 pr 2015 bstract Here I show compatibility of two definition of generalized curvature bounds the lower bound for sectional
More informationLogarithmic Sobolev Inequalities
Logarithmic Sobolev Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France logarithmic Sobolev inequalities what they are, some history analytic, geometric, optimal transportation proofs
More informationSpaces with Ricci curvature bounded from below
Spaces with Ricci curvature bounded from below Nicola Gigli February 23, 2015 Topics 1) On the definition of spaces with Ricci curvature bounded from below 2) Analytic properties of RCD(K, N) spaces 3)
More informationLocal semiconvexity of Kantorovich potentials on non-compact manifolds
Local semiconvexity of Kantorovich potentials on non-compact manifolds Alessio Figalli, Nicola Gigli Abstract We prove that any Kantorovich potential for the cost function c = d / on a Riemannian manifold
More informationRESEARCH STATEMENT MICHAEL MUNN
RESEARCH STATEMENT MICHAEL MUNN Ricci curvature plays an important role in understanding the relationship between the geometry and topology of Riemannian manifolds. Perhaps the most notable results in
More informationSpaces with Ricci curvature bounded from below
Spaces with Ricci curvature bounded from below Nicola Gigli March 10, 2014 Lessons Basics of optimal transport Definition of spaces with Ricci curvature bounded from below Analysis on spaces with Ricci
More informationarxiv:math/ v3 [math.dg] 30 Jul 2007
OPTIMAL TRANSPORT AND RICCI CURVATURE FOR METRIC-MEASURE SPACES ariv:math/0610154v3 [math.dg] 30 Jul 2007 JOHN LOTT Abstract. We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for
More informationRicci curvature for metric-measure spaces via optimal transport
Annals of athematics, 169 (2009), 903 991 Ricci curvature for metric-measure spaces via optimal transport By John Lott and Cédric Villani* Abstract We define a notion of a measured length space having
More informationarxiv: v4 [math.dg] 18 Jun 2015
SMOOTHING 3-DIMENSIONAL POLYHEDRAL SPACES NINA LEBEDEVA, VLADIMIR MATVEEV, ANTON PETRUNIN, AND VSEVOLOD SHEVCHISHIN arxiv:1411.0307v4 [math.dg] 18 Jun 2015 Abstract. We show that 3-dimensional polyhedral
More informationDiscrete Ricci curvature via convexity of the entropy
Discrete Ricci curvature via convexity of the entropy Jan Maas University of Bonn Joint work with Matthias Erbar Simons Institute for the Theory of Computing UC Berkeley 2 October 2013 Starting point McCann
More informationWEAK CURVATURE CONDITIONS AND FUNCTIONAL INEQUALITIES
WEA CURVATURE CODITIOS AD FUCTIOAL IEQUALITIES JOH LOTT AD CÉDRIC VILLAI Abstract. We give sufficient conditions for a measured length space (, d, ν to admit local and global Poincaré inequalities, along
More informationGeneralized Ricci Bounds and Convergence of Metric Measure Spaces
Generalized Ricci Bounds and Convergence of Metric Measure Spaces Bornes Généralisées de la Courbure Ricci et Convergence des Espaces Métriques Mesurés Karl-Theodor Sturm a a Institut für Angewandte Mathematik,
More informationAn upper bound for curvature integral
An upper bound for curvature integral Anton Petrunin Abstract Here I show that the integral of scalar curvature of a closed Riemannian manifold can be bounded from above in terms of its dimension, diameter,
More informationFokker-Planck Equation on Graph with Finite Vertices
Fokker-Planck Equation on Graph with Finite Vertices January 13, 2011 Jointly with S-N Chow (Georgia Tech) Wen Huang (USTC) Hao-min Zhou(Georgia Tech) Functional Inequalities and Discrete Spaces Outline
More informationStrictly convex functions on complete Finsler manifolds
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 4, November 2016, pp. 623 627. DOI 10.1007/s12044-016-0307-2 Strictly convex functions on complete Finsler manifolds YOE ITOKAWA 1, KATSUHIRO SHIOHAMA
More informationA few words about the MTW tensor
A few words about the Ma-Trudinger-Wang tensor Université Nice - Sophia Antipolis & Institut Universitaire de France Salah Baouendi Memorial Conference (Tunis, March 2014) The Ma-Trudinger-Wang tensor
More informationChapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves
Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction
More informationHOMEWORK 2 - RIEMANNIAN GEOMETRY. 1. Problems In what follows (M, g) will always denote a Riemannian manifold with a Levi-Civita connection.
HOMEWORK 2 - RIEMANNIAN GEOMETRY ANDRÉ NEVES 1. Problems In what follows (M, g will always denote a Riemannian manifold with a Levi-Civita connection. 1 Let X, Y, Z be vector fields on M so that X(p Z(p
More informationDisplacement convexity of the relative entropy in the discrete h
Displacement convexity of the relative entropy in the discrete hypercube LAMA Université Paris Est Marne-la-Vallée Phenomena in high dimensions in geometric analysis, random matrices, and computational
More informationMoment Measures. Bo az Klartag. Tel Aviv University. Talk at the asymptotic geometric analysis seminar. Tel Aviv, May 2013
Tel Aviv University Talk at the asymptotic geometric analysis seminar Tel Aviv, May 2013 Joint work with Dario Cordero-Erausquin. A bijection We present a correspondence between convex functions and Borel
More informationarxiv: v1 [math.mg] 28 Sep 2017
Ricci tensor on smooth metric measure space with boundary Bang-Xian Han October 2, 2017 arxiv:1709.10143v1 [math.mg] 28 Sep 2017 Abstract Theaim of this note is to studythemeasure-valued Ricci tensor on
More informationRicci Curvature and Bochner Formula on Alexandrov Spaces
Ricci Curvature and Bochner Formula on Alexandrov Spaces Sun Yat-sen University March 18, 2013 (work with Prof. Xi-Ping Zhu) Contents Alexandrov Spaces Generalized Ricci Curvature Geometric and Analytic
More informationEXAMPLE OF A FIRST ORDER DISPLACEMENT CONVEX FUNCTIONAL
EXAMPLE OF A FIRST ORDER DISPLACEMENT CONVEX FUNCTIONAL JOSÉ A. CARRILLO AND DEJAN SLEPČEV Abstract. We present a family of first-order functionals which are displacement convex, that is convex along the
More informationNOTE ON ASYMPTOTICALLY CONICAL EXPANDING RICCI SOLITONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 NOTE ON ASYMPTOTICALLY CONICAL EXPANDING RICCI SOLITONS JOHN LOTT AND PATRICK WILSON (Communicated
More informationVOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE
VOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE KRISTOPHER TAPP Abstract. The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between
More informationFrom the Brunn-Minkowski inequality to a class of Poincaré type inequalities
arxiv:math/0703584v1 [math.fa] 20 Mar 2007 From the Brunn-Minkowski inequality to a class of Poincaré type inequalities Andrea Colesanti Abstract We present an argument which leads from the Brunn-Minkowski
More informationMetric measure spaces with Riemannian Ricci curvature bounded from below Lecture I
1 Metric measure spaces with Riemannian Ricci curvature bounded from below Lecture I Giuseppe Savaré http://www.imati.cnr.it/ savare Dipartimento di Matematica, Università di Pavia Analysis and Geometry
More informationDiscrete transport problems and the concavity of entropy
Discrete transport problems and the concavity of entropy Bristol Probability and Statistics Seminar, March 2014 Funded by EPSRC Information Geometry of Graphs EP/I009450/1 Paper arxiv:1303.3381 Motivating
More informationcurvature, mixing, and entropic interpolation Simons Feb-2016 and CSE 599s Lecture 13
curvature, mixing, and entropic interpolation Simons Feb-2016 and CSE 599s Lecture 13 James R. Lee University of Washington Joint with Ronen Eldan (Weizmann) and Joseph Lehec (Paris-Dauphine) Markov chain
More informationHeat flow on Alexandrov spaces
Heat flow on Alexandrov spaces Nicola Gigli, Kazumasa Kuwada, Shin-ichi Ohta January 19, 2012 Abstract We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet
More informationThe volume growth of complete gradient shrinking Ricci solitons
arxiv:0904.0798v [math.dg] Apr 009 The volume growth of complete gradient shrinking Ricci solitons Ovidiu Munteanu Abstract We prove that any gradient shrinking Ricci soliton has at most Euclidean volume
More informationTHE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS
THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic
More informationCurvature and the continuity of optimal transportation maps
Curvature and the continuity of optimal transportation maps Young-Heon Kim and Robert J. McCann Department of Mathematics, University of Toronto June 23, 2007 Monge-Kantorovitch Problem Mass Transportation
More informationICM 2014: The Structure and Meaning. of Ricci Curvature. Aaron Naber ICM 2014: Aaron Naber
Outline of Talk Background and Limit Spaces Structure of Spaces with Lower Ricci Regularity of Spaces with Bounded Ricci Characterizing Ricci Background: s (M n, g, x) n-dimensional pointed Riemannian
More informationResults from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2000
2000k:53038 53C23 20F65 53C70 57M07 Bridson, Martin R. (4-OX); Haefliger, André (CH-GENV-SM) Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles
More informationMass transportation methods in functional inequalities and a new family of sharp constrained Sobolev inequalities
Mass transportation methods in functional inequalities and a new family of sharp constrained Sobolev inequalities Robin Neumayer Abstract In recent decades, developments in the theory of mass transportation
More informationHeat Flows, Geometric and Functional Inequalities
Heat Flows, Geometric and Functional Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France heat flow and semigroup interpolations Duhamel formula (19th century) pde, probability, dynamics
More informationRough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow
Rough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow Lashi Bandara Mathematical Sciences Chalmers University of Technology and University of Gothenburg 22
More informationDiscrete Ricci curvature: Open problems
Discrete Ricci curvature: Open problems Yann Ollivier, May 2008 Abstract This document lists some open problems related to the notion of discrete Ricci curvature defined in [Oll09, Oll07]. Do not hesitate
More informationON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES
ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES TAKUMI YOKOTA Abstract. It was shown by F. Wilhelm that Gromov s filling radius of any positively curved closed Riemannian manifolds are less
More informationWASSERSTEIN GEOMETRY OF GAUSSIAN MEASURES
WASSERSTEIN GEOMETRY OF GAUSSIAN MEASURES ASUKA TAKATSU Abstract. This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L 2 -Wasserstein geometry. The
More informationOn the Intrinsic Differentiability Theorem of Gromov-Schoen
On the Intrinsic Differentiability Theorem of Gromov-Schoen Georgios Daskalopoulos Brown University daskal@math.brown.edu Chikako Mese 2 Johns Hopkins University cmese@math.jhu.edu Abstract In this note,
More informationON THE VOLUME MEASURE OF NON-SMOOTH SPACES WITH RICCI CURVATURE BOUNDED BELOW
ON THE VOLUME MEASURE OF NON-SMOOTH SPACES WITH RICCI CURVATURE BOUNDED BELOW MARTIN KELL AND ANDREA MONDINO Abstract. We prove that, given an RCD (K, N)-space (X, d, m), then it is possible to m-essentially
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationMetric and comparison geometry
Surveys in Differential Geometry XI Metric and comparison geometry Jeff Cheeger and Karsten Grove The present volume surveys some of the important recent developments in metric geometry and comparison
More informationIntegral Jensen inequality
Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a
More information1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
More informationRicci curvature bounds for warped products and cones
Ricci curvature bounds for warped products and cones Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch Naturwissenschaftlichen Fakultät der Rheinischen Friedrich Wilhelms Universität
More informationMetric measure spaces with Ricci lower bounds, Lecture 1
Metric measure spaces with Ricci lower bounds, Lecture 1 Andrea Mondino (Zurich University) MSRI-Berkeley 20 th January 2016 Motivation-Smooth setting:comparison geometry Question: (M, g) smooth Riemannian
More informationA NOTION OF NONPOSITIVE CURVATURE FOR GENERAL METRIC SPACES
A NOTION OF NONPOSITIVE CURVATURE FOR GENERAL METRIC SPACES MIROSLAV BAČÁK, BOBO HUA, JÜRGEN JOST, MARTIN KELL, AND ARMIN SCHIKORRA Abstract. We introduce a new definition of nonpositive curvature in metric
More information4.7 The Levi-Civita connection and parallel transport
Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves
More informationarxiv: v2 [math.ap] 23 Apr 2014
Multi-marginal Monge-Kantorovich transport problems: A characterization of solutions arxiv:1403.3389v2 [math.ap] 23 Apr 2014 Abbas Moameni Department of Mathematics and Computer Science, University of
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationON TWO-DIMENSIONAL MINIMAL FILLINGS. S. V. Ivanov
ON TWO-DIMENSIONAL MINIMAL FILLINGS S. V. Ivanov Abstract. We consider Riemannian metrics in the two-dimensional disk D (with boundary). We prove that, if a metric g 0 is such that every two interior points
More informationOptimal transport and Ricci curvature in Finsler geometry
ASP.tex : 2010/1/22 (15:59) page: 1 Advanced Studies in Pure athematics, pp. 1 20 Optimal transport and Ricci curvature in Finsler geometry Abstract. Shin-ichi Ohta This is a survey article on recent progress
More informationRiemannian geometry of surfaces
Riemannian geometry of surfaces In this note, we will learn how to make sense of the concepts of differential geometry on a surface M, which is not necessarily situated in R 3. This intrinsic approach
More informationDynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Was
Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Wasserstein Space With many discussions with Yann Brenier and Wilfrid Gangbo Brenierfest, IHP, January 9-13, 2017 ain points of the
More informationOn the regularity of solutions of optimal transportation problems
On the regularity of solutions of optimal transportation problems Grégoire Loeper April 25, 2008 Abstract We give a necessary and sufficient condition on the cost function so that the map solution of Monge
More informationA REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH SPACE
Theory of Stochastic Processes Vol. 21 (37), no. 2, 2016, pp. 84 90 G. V. RIABOV A REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH
More informationGeneralized Orlicz spaces and Wasserstein distances for convex concave scale functions
Bull. Sci. math. 135 (2011 795 802 www.elsevier.com/locate/bulsci Generalized Orlicz spaces and Wasserstein distances for convex concave scale functions Karl-Theodor Sturm Institut für Angewandte Mathematik,
More informationAn Alexandroff Bakelman Pucci estimate on Riemannian manifolds
Available online at www.sciencedirect.com Advances in Mathematics 232 (2013) 499 512 www.elsevier.com/locate/aim An Alexandroff Bakelman Pucci estimate on Riemannian manifolds Yu Wang, Xiangwen Zhang Department
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More informationCalderón-Zygmund inequality on noncompact Riem. manifolds
The Calderón-Zygmund inequality on noncompact Riemannian manifolds Institut für Mathematik Humboldt-Universität zu Berlin Geometric Structures and Spectral Invariants Berlin, May 16, 2014 This talk is
More informationthe canonical shrinking soliton associated to a ricci flow
the canonical shrinking soliton associated to a ricci flow Esther Cabezas-Rivas and Peter. Topping 15 April 2009 Abstract To every Ricci flow on a manifold over a time interval I R, we associate a shrinking
More informationL -uniqueness of Schrödinger operators on a Riemannian manifold
L -uniqueness of Schrödinger operators on a Riemannian manifold Ludovic Dan Lemle Abstract. The main purpose of this paper is to study L -uniqueness of Schrödinger operators and generalized Schrödinger
More informationAN OPTIMAL TRANSPORT VIEW ON SCHRÖDINGER S EQUATION
AN OPTIAL TRANSPORT VIEW ON SCHRÖDINGER S EQUATION ax-k. von Renesse Abstract We show that the Schrödinger equation is a lift of Newton s third law of motion Ẇ µ µ = W F (µ) on the space of probability
More informationTransport Continuity Property
On Riemannian manifolds satisfying the Transport Continuity Property Université de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) I. Statement of the problem Optimal transport on Riemannian
More informationCOMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES. 1. Introduction
COMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES ANTONIO CAÑETE AND CÉSAR ROSALES Abstract. We consider a smooth Euclidean solid cone endowed with a smooth
More informationREGULARITY OF MONOTONE TRANSPORT MAPS BETWEEN UNBOUNDED DOMAINS
REGULARITY OF MONOTONE TRANSPORT MAPS BETWEEN UNBOUNDED DOMAINS DARIO CORDERO-ERAUSQUIN AND ALESSIO FIGALLI A Luis A. Caffarelli en su 70 años, con amistad y admiración Abstract. The regularity of monotone
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationarxiv: v1 [math.mg] 13 Nov 2017
A COUNTEREXAMPLE TO GLUING THEOREMS FOR MCP METRIC MEASURE SPACES LUCA RIZZI arxiv:1711.04499v1 [math.mg] 13 Nov 2017 Abstract. Perelman s doubling theorem asserts that the metric space obtained by gluing
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationPoisson Equation on Closed Manifolds
Poisson Equation on Closed anifolds Andrew acdougall December 15, 2011 1 Introduction The purpose of this project is to investigate the poisson equation φ = ρ on closed manifolds (compact manifolds without
More informationON THE VOLUME MEASURE OF NON-SMOOTH SPACES WITH RICCI CURVATURE BOUNDED BELOW
ON THE VOLUME MEASURE OF NON-SMOOTH SPACES WITH RICCI CURVATURE BOUNDED BELOW MARTIN KELL AND ANDREA MONDINO Abstract. We prove that, given an RCD (K, N)-space (X, d, m), then it is possible to m-essentially
More informationStable minimal cones in R 8 and R 9 with constant scalar curvature
Revista Colombiana de Matemáticas Volumen 6 (2002), páginas 97 106 Stable minimal cones in R 8 and R 9 with constant scalar curvature Oscar Perdomo* Universidad del Valle, Cali, COLOMBIA Abstract. In this
More informationON THE GROUND STATE OF QUANTUM LAYERS
ON THE GROUND STATE OF QUANTUM LAYERS ZHIQIN LU 1. Introduction The problem is from mesoscopic physics: let p : R 3 be an embedded surface in R 3, we assume that (1) is orientable, complete, but non-compact;
More informationb i (µ, x, s) ei ϕ(x) µ s (dx) ds (2) i=1
NONLINEAR EVOLTION EQATIONS FOR MEASRES ON INFINITE DIMENSIONAL SPACES V.I. Bogachev 1, G. Da Prato 2, M. Röckner 3, S.V. Shaposhnikov 1 The goal of this work is to prove the existence of a solution to
More informationSemiconcave functions in Alexandrov s geometry
Semiconcave functions in Alexandrov s geometry Anton Petrunin Abstract The following is a compilation of some techniques in Alexandrov s geometry which are directly connected to convexity. 0 Introduction
More informationShape Matching: A Metric Geometry Approach. Facundo Mémoli. CS 468, Stanford University, Fall 2008.
Shape Matching: A Metric Geometry Approach Facundo Mémoli. CS 468, Stanford University, Fall 2008. 1 General stuff: class attendance is mandatory will post possible projects soon a couple of days prior
More informationThe Ricci Flow Approach to 3-Manifold Topology. John Lott
The Ricci Flow Approach to 3-Manifold Topology John Lott Two-dimensional topology All of the compact surfaces that anyone has ever seen : These are all of the compact connected oriented surfaces without
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationDIFFERENTIAL GEOMETRY. LECTURE 12-13,
DIFFERENTIAL GEOMETRY. LECTURE 12-13, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of
More informationFinite Presentations of Hyperbolic Groups
Finite Presentations of Hyperbolic Groups Joseph Wells Arizona State University May, 204 Groups into Metric Spaces Metric spaces and the geodesics therein are absolutely foundational to geometry. The central
More informationConvexity in R n. The following lemma will be needed in a while. Lemma 1 Let x E, u R n. If τ I(x, u), τ 0, define. f(x + τu) f(x). τ.
Convexity in R n Let E be a convex subset of R n. A function f : E (, ] is convex iff f(tx + (1 t)y) (1 t)f(x) + tf(y) x, y E, t [0, 1]. A similar definition holds in any vector space. A topology is needed
More informationOPTIMAL TRANSPORT AND CURVATURE
OPTIAL TRANSPORT AND CURVATURE ALESSIO FIGALLI AND CÉDRIC VILLANI Introduction These notes record the six lectures for the CIE Summer Course held by the second author in Cetraro during the week of June
More informationAPPROXIMATING COARSE RICCI CURVATURE ON METRIC MEASURE SPACES WITH APPLICATIONS TO SUBMANIFOLDS OF EUCLIDEAN SPACE
APPROXIMATING COARSE RICCI CURVATURE ON METRIC MEASURE SPACES WITH APPLICATIONS TO SUBMANIFOLDS OF EUCLIDEAN SPACE ANTONIO G. ACHE AND MICAH W. WARREN Abstract. For a submanifold Σ R N Belkin and Niyogi
More informationDisplacement convexity of generalized relative entropies. II
Displacement convexity of generalized relative entropies. II Shin-ichi Ohta and Asuka Takatsu arch 3, 23 Abstract We introduce a class of generalized relative entropies inspired by the Bregman divergence
More informationAN INTRODUCTION TO VARIFOLDS DANIEL WESER. These notes are from a talk given in the Junior Analysis seminar at UT Austin on April 27th, 2018.
AN INTRODUCTION TO VARIFOLDS DANIEL WESER These notes are from a talk given in the Junior Analysis seminar at UT Austin on April 27th, 2018. 1 Introduction Why varifolds? A varifold is a measure-theoretic
More informationThe Geometrization Theorem
The Geometrization Theorem Matthew D. Brown Wednesday, December 19, 2012 In this paper, we discuss the Geometrization Theorem, formerly Thurston s Geometrization Conjecture, which is essentially the statement
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationDefinitions of curvature bounded below
Chapter 6 Definitions of curvature bounded below In section 6.1, we will start with a global definition of Alexandrov spaces via (1+3)-point comparison. In section 6.2, we give a number of equivalent angle
More informationSYNTHETIC GEOMETRY AND GENERALISED FUNCTIONS Introduction. Novi Sad J. Math. Vol. 41, No. 1, 2011, James D.E. Grant 2
Novi Sad J. Math. Vol. 41, No. 1, 2011, 75-84 SYNTHETIC GEOMETRY AND GENERALISED FUNCTIONS 1 James D.E. Grant 2 Abstract. We review some aspects of the geometry of length spaces and metric spaces, in particular
More informationδ-hyperbolic SPACES SIDDHARTHA GADGIL
δ-hyperbolic SPACES SIDDHARTHA GADGIL Abstract. These are notes for the Chennai TMGT conference on δ-hyperbolic spaces corresponding to chapter III.H in the book of Bridson and Haefliger. When viewed from
More informationEntropic curvature-dimension condition and Bochner s inequality
Entropic curvature-dimension condition and Bochner s inequality Kazumasa Kuwada (Ochanomizu University) joint work with M. Erbar and K.-Th. Sturm (Univ. Bonn) German-Japanese conference on stochastic analysis
More informationSome topics in sub-riemannian geometry
Some topics in sub-riemannian geometry Luca Rizzi CNRS, Institut Fourier Mathematical Colloquium Universität Bern - December 19 2016 Sub-Riemannian geometry Known under many names: Carnot-Carathéodory
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationWasserstein GAN. Juho Lee. Jan 23, 2017
Wasserstein GAN Juho Lee Jan 23, 2017 Wasserstein GAN (WGAN) Arxiv submission Martin Arjovsky, Soumith Chintala, and Léon Bottou A new GAN model minimizing the Earth-Mover s distance (Wasserstein-1 distance)
More informationLECTURE 16: CONJUGATE AND CUT POINTS
LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M, g) be Riemannian and γ : [a, b] M a geodesic. Then by definition, exp p ((t a) γ(a)) = γ(t). We know that exp p is a diffeomorphism near
More informationThe Lusin Theorem and Horizontal Graphs in the Heisenberg Group
Analysis and Geometry in Metric Spaces Research Article DOI: 10.2478/agms-2013-0008 AGMS 2013 295-301 The Lusin Theorem and Horizontal Graphs in the Heisenberg Group Abstract In this paper we prove that
More informationOn a Class of Multidimensional Optimal Transportation Problems
Journal of Convex Analysis Volume 10 (2003), No. 2, 517 529 On a Class of Multidimensional Optimal Transportation Problems G. Carlier Université Bordeaux 1, MAB, UMR CNRS 5466, France and Université Bordeaux
More information