{"id":1637,"date":"2010-03-16T09:09:55","date_gmt":"2010-03-16T14:09:55","guid":{"rendered":"http:\/\/sites.williams.edu\/Morgan\/?p=1637"},"modified":"2022-12-23T06:31:26","modified_gmt":"2022-12-23T11:31:26","slug":"manifolds-with-density-fuller-references","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2010\/03\/16\/manifolds-with-density-fuller-references\/","title":{"rendered":"Manifolds with Density: Fuller References"},"content":{"rendered":"

SELECTED PUBLICATIONS IN THE HISTORY OF MANIFOLDS WITH DENSITY<\/a>:<\/strong><\/p>\n

[1959] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Indo. Control 2 (1959), 101-112, Eqn. 2.3. Gives a version of Gaussian log-Sobolev inequality, used by Perelman, often attributed to Gross [1975] or sometimes Federbush [1969].<\/p>\n

[1966] E. Nelson, A quartic interaction in two dimensions, mathematical Theory of Elementary Particles (Goodman, R. and Segal, I., eds.), MIT Press, 1966, 69-73. Gross [G] says that the entire subject of logarithmic Sobolev inequalities and contractivity properties of semigroups was started in this paper.<\/p>\n

[1966] Harper, L. H. Optimal numberings and isoperimetric problems on graphs. J. Combinatorial Theory 1 1966 385-393. Apparently uses measure and metric, cited by Ledoux-Talagrand [1991], both cited by [Ros, \u00a71.4, p. 182].<\/p>\n

[1969] Paul Federbush, A partially alternate derivation of a result of Nelson, J. Math. Phys. 10 (1969), 50-52. Gives Gaussian log-Sobolev inequality, used by Perelman, often attributed to Gross [1975], actually probably due to Stam [1959].<\/p>\n

[1970] Andr\u00e9 Lichnerowicz, Vari\u00e9t\u00e9s riemanniennes a tenseur C non n\u00e9gatif, C. R. Acad. Sci. Paris S\u00e9r. A-B 271 (1970), A650-A653. Studies Ric – Hess log density to prove splitting theorems.<\/p>\n

[1973] E Nelson, The free Markov field, J. Funct. Analy. 12, 211-227. Gross survey ([G] below) says equivalent form of Gaussian log-Sobolev inequality.<\/p>\n

[1975] Christer Borell, The Brunn-Minkowski inequality in Gauss Space, Invent. Math. 30 (1975) 207-216. Also: V. N. Sudakov and B. S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures, J. Soviet Math. (1978), 9-18 (1974 in Russian). Proof of Gaussian isoperimetric inequality.<\/p>\n

[1975] Gross, Leonard, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083.\u00a0 Proves Gaussian log-Sobolev inequality, used by Perelman. Not yet realized that it was trivial consequence of Gaussian isoperimetric inequality via analytic version which [Ros,\u00a0 \u00a73.4] traces back to Ehrhard [E, 1984] and Bobkov [B7, 1997], first observed by Ledoux ’94 and Beckner ’96 (published [1999]) (see Morgan Blog<\/a> and email from Milman). Precursors are Stam [1959] and Federbush [1969] (see Gross survey [G]).<\/p>\n

[1977] R. R. Coifman and G. Weiss, Extensions of Hardy Spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. First appearance (as I learned from Cheeger) of metric-measure spaces (singular manifolds\u00a0 with density) called “homogeneous spaces.”<\/p>\n

[1981]\u00a0\u00a0M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkh\u00e4user Classics, Birkh\u00e4user Boston Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. Metric measure spaces; see Shioya below.<\/p>\n

[1982] Edward Witten. Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661–692 (1983). Pointed out to me by Doan.\u00a0 d_t = e^-ht d e^ht \u00a0 The main idea seems to be the same as Perelman’s: changing the density does not change the topology but it can help with the analysis.<\/p>\n

[1984] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schr\u00f6dinger operators and Dirichlet Laplacians, J. Funct. Anal., 59 (1984) 335-395.<\/p>\n

[1985] D. Bakry and M. \u00c9mery, Diffusions Hypercontractive, S\u00e9minaire de Probabilit\u00e9s XIX 1983\/4, Lecture Notes Math. 1123, Springer, 1985, 177-206. Generalized Ricci curvatures. They mention the earlier classical Lichnerowicz-Bochner-Weitzenb\u00f6ck formula right after their statement of their Proposition 3, as pointed out to me by Houdr\u00e9.<\/p>\n

[1989] Kenji Fukaya. Collapsing Riemannian manifolds to ones with lower dimension. I, II. J. Math. Soc. Japan 41 (1989), 333\u2013356. Preceding Cheeger and Colding, 1997.<\/p>\n

[1991] Michel Ledoux and Michel Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. Springer-Verlag, New York, 1991. First text on topic.<\/p>\n

[1991] Isaac Chavel and Edgar A. Feldman, Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds,\u00a0Duke Math. J. 64 (1991), 473-499. Modified isoperimetric and heat diffusion, weighted Laplacian. “…we introduce weighted Riemannian spaces and graphs, for we will require this expanded category to prove Theorem 2-4.” p. 478.<\/p>\n

[1996] D. Bakry and M. Ledoux, L\u00e9vy-Gromov isoperimetric inequality for an infinite-dimensional diffusion generator, Inv. Math. 123 (1996), 259-281.<\/p>\n

[1997] Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. I, II, III,\u00a0 J. Differential Geom. 46 (1997), 406-480, 54 (2000), 13-35, 37-74.<\/p>\n

[1999] William Beckner, Geometric asymptotics and the logarithmic Sobolev inequality. Forum Math. 11 (1999), no. 1, 105-137.<\/p>\n

[2000] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related\u00a0inequalities, Comm. Math. Phys. 214:3, (2000), 547\u2013563. Additional density on Gauss space log concave implies volume-preserving Lipschitz image of Gauss (implies isoperimetrically worse than Gauss, hence Levy-Gromov for densities on Euclidean space).<\/p>\n

[2003] Misha Gromov, Isoperimetry of waists and concentration of maps, Geom. Func. Anal. 13 (2003), 178-215.<\/p>\n

[2003] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org (2003). Conceptual development via manifolds with density; specific use of Gross’s Gaussian log-Sobolev inequality. Sect 3.3:\u00a0 “Logarithmic Sobolev inequalities is a vast area of research; see [G] for a survey and bibliography up to the year 1992; the influence of the curvature was discussed by Bakry-Emery [B-Em]. In the context of geometric evolution equations, the logarithmic Sobolev inequality occurs in Ecker [E1].”<\/p>\n

[2004] Vincent Bayle, Propri\u00e9t\u00e9s de concavit\u00e9 du profil isop\u00e9rim\u00e9trique et applicationes, graduate thesis, Institut Fourier, Universite Joseph-Fourier, Grenoble I, 2004. Beautiful survey plus second variation version of Bakry-Emery and more.<\/p>\n

[2005] Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858. Survey and simple derivation of weighted Levy-Gromov with uniqueness. Revised as Chapter 18 of 4th edition of Geometric Measure Theory: a Beginner’s Guide [2009].<\/p>\n

[2006] Alexander Grigor’yan, Heat kernels on weighted manifolds\u00a0and applications. The ubiquitous heat kernel, 93-191, Contemp. Math., 398, Amer. Math. Soc., Providence, RI, 2006. Credits Chavel-Feldman 1991 and Davies 1992 for weighted Laplacian.<\/p>\n

“The notion of a weighted Laplacian was introduced by I. Chavel and E. Feldman [1991] and by E. B. Davies [D, 1992]. Many facts from the analysis on weighted manifolds are similar to those on Riemannian manifolds. However, in the former setting one has an added flexibility of changing the measure without changing the underlying Riemannian structure, which happens to be a powerful technical tool, as was earlier observed by E. B. Davies and B. Simon [1984]. A natural setup for this approach would be a metric measure space with an energy form in the spirit of [FOT, 1994], but this would bring additional technical complications, caused by the singularity of the\u00a0space.” Ex. 2.1. Scaling metric by a(x) and measure by b(x) yields new Laplacian (1\/b)div((b\/a)grad). 2.4. Laplacian on model manifolds (surfaces with density of revolution). Lap\u00b5<\/sub> = \u22022<\/sup>\/\u2202r2<\/sup> + m(r) \u2202\/\u2202r + \u03c8-2<\/sup> Lap_theta m(r) = (log S(r))’ = (n-1)(log g(r))’ + 2\u03c8’.<\/p>\n

[2006] Karl-Theodor Sturm. On the geometry of metric measure spaces, I, II. Acta Math. 196 (2006), 65-131 and 133-177.<\/p>\n

[2006] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009), 903\u2013991; arXiv.org (2006).<\/p>\n

[2006] Sun-Yung A. Chang, Matthew J. Gursky, and Paul Yang, Conformal invariants associated to a measure, Proc. Natl. Acad. Sci. USA 103 (2006), 2535-2540.<\/p>\n

[2007] Guofang Wei and Will Wylie, Comparison geometry for the Bakry-\u00c9mery Ricci tensor, arXiv.org (2007). And see references therein.<\/p>\n

[2009] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide. Academic Press, fourth edition, 2009.<\/p>\n

RECENT:<\/strong><\/p>\n

[2007] Peter W. Michor and David Mumford. An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal., 23(1):74\u2013113, 2007.<\/p>\n

[2008] Edward M.\u00a0Fan,\u00a0Topology of three-manifolds with positive P<\/em>-scalar curvature.\u00a0Proc. Amer. Math. Soc. 136 (2008), 3255\u20133261. [2009] Ruan, Qi-hua, Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 6, 71\u201374.<\/p>\n

Summary: In this paper, we generalize Cheng’s maximal diameter theorem and Bishop’s volume comparison theorem to manifolds with an m-Bakry-\u00c9mery Ricci curvature. As applications, we obtain some rigidity theorems on the warped product.”<\/p>\n

[2009] Y. Ollivier, Ricci Curvature of Markov Chains on Metric Spaces, J. Funct. Anal. 256 (2009), 810-864. http:\/\/arxiv.org\/abs\/math\/0701886<\/p>\n

[2010] Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons [constant positive generalized Ricci curvature], J. Diff. Geom. 85 (2010), 175-185. Generalized Bishop volume comparison theorem. Cites Wei-Wylie 2007.<\/p>\n

[2010] Ma, Li; Du, Sheng-Hua, Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. C. R. Math. Acad. Sci. Paris 348 (2010), 1203\u20131206.<\/p>\n

[2010]\u00a0Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran, Isoperimetric regions in the plane with density r^p, New York J. Math. 16 (2010), 31-51. http:\/\/nyjm.albany.edu\/j\/2010\/16-4.html<\/p>\n

[2010] Kolesnikov and Zhdanov, On isoperimetric sets of radially symmetric measures, arXiv.org, 2010. Corollary 6.17. If V >= 0 and convex, then e^V dx satisfies the classical isoperimetric inequality. And much more.<\/p>\n

[2010]\u00a0Pak Tung Ho,The structure of \u03d5-stable minimal hypersurfaces in manifolds of nonnegative P<\/em>-scalar curvature.\u00a0Math. Ann. 348 (2010), 319\u2013332.<\/p>\n

[2010] Emil Saucan, Curvature based triangulation of metric measure spaces. http:\/\/arxiv.org\/abs\/1002.0007v1<\/p>\n

[2010] Kevin Brighton, A Liouville-type theorem for smooth metric measure spaces. http:\/\/arxiv.org\/abs\/1006.0751<\/p>\n

[2011]\u00a0Doan The Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys. 61 (2011),\u00a01625-1629.<\/p>\n

[2011] Ovidiu Munteanu and Jiaping Wang, Smooth metric measure spaces with non-negative curvature, arXiv.org, 2011. “As an application, we conclude steady Ricci solitons must be connected at infinity.” http:\/\/arxiv.org\/abs\/1103.0746<\/p>\n

[2011] Emil Saucan, A simple sampling method for metric measure spaces. http:\/\/arxiv.org\/abs\/1103.3843\u00a0 Cites Corwin et al. as well as my MwD.<\/p>\n

[2011] J\u00fcrgen Jost and Shiping Liu, Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs. http:\/\/arxiv.org\/abs\/1103.4037<\/p>\n

[2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savar\u00e9, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, http:\/\/arxiv.org\/abs\/1106.2090<\/p>\n

[2011] Sean Howe, The Log-Convex Density Conjecture and vertical surface area in warped products, preprint (2011), http:\/\/arxiv.org\/abs\/1107.4402.<\/p>\n

[2011] Tapio Rajala, Local Poincar\u00e9 inequalities from stable curvature conditions on metric spaces. http:\/\/arXiv.org\/abs\/1107.4842v1<\/p>\n

[2011] David Bate and Gareth Speight, Differentiability, porosity and doubling in metric measure spaces, http:\/\/arxiv.org\/abs\/1108.0318<\/p>\n

[2011]\u00a0Emanuel Milman,\u00a0Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition,\u00a0http:\/\/arxiv.org\/abs\/1108.4609v3,\u00a0J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041\u20131078. [2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savar\u00e9, Metric measure spaces with Riemannian Ricci curvature bounded from below, http:\/\/arxiv.org\/abs\/1109.0222<\/p>\n

[2011] Martin Bauer, Philipp Harms, and Peter W. Michor, Sobolev metrics on shape space II: weighted sobolev metrics and almost local metries, arXiv:1109.0404v1 [math.DG] 2 Sep 2011. See also references therein, including [2007, Michor and Mumford] above.<\/p>\n

[2011] Jasun Gong, Differentiability of Lipschitz functions on doubling metric measure spaces, http:\/\/arxiv.org\/abs\/1110.4279.<\/p>\n

[2011] Siddhartha Gadgi and Manjunath Krishnapur, Lipschitz correspondence between metric measure spaces and random distance matrices, http:\/\/arxiv.org\/abs\/1110.6333.<\/p>\n

[2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savar\u00e9, Density of Lipschitz functons and equivalence of weak gradients in metric measure spaces, http:\/\/arxiv.org\/abs\/1111.3730<\/p>\n

[2011] Ben Andrews and Lei Ni, Eignvalue comparison on Bakry-Emery manifolds, http:\/\/arxiv.org\/abs\/1111.4967<\/p>\n

[2011] Tapio Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditons of Sturm, http:\/\/arxiv.org\/abs\/1111.5526<\/p>\n

[2011] Wolfang L\u00f6hr, Equivalence of Gromov-Prohorov- and Gromov’s box-metric on the space of metric measure spaces, http:\/\/arxiv.org\/abs\/1111.5837<\/p>\n

[2011] Shouhei Honda, A weak second differentiable structure on rectifiable metric measure spaces, http:\/\/arxiv.org\/abs\/1112.0099<\/p>\n

[2012]\u00a0Alexander D\u00edaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589\u2013619;\u00a0arXiv.org<\/a>\u00a0(2010); see blog posts\u00a01<\/a>\u00a0and\u00a02<\/a>.<\/p>\n

[2012] Yan-Hui Su and Hui-Chun Zhang. Rigidity of manifolds with Bakry-Emery Ricci curvature bounded below. Geometriae Dedicata 160 (2012), 1\u201311.<\/p>\n

[2012] Luigi Ambrosio, Nicola Gigli, Giuseppe Savar\u00e9, Heat flow and calculus on metric measure spaces with Ricci curvature bounded below – the compact case, http:\/\/arxiv.org\/abs\/1205.3288<\/p>\n

[2012] Nicola Gigli, On the differential structure of metric measure spaces and applications, http:\/\/arxiv.org\/abs\/1205.6622<\/p>\n

[2012] Luigi Ambrosio, Nicola Gigli, Andrea Mondino, Tapio Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with sigma-finite measure, http:\/\/arxiv.org\/abs\/1207.4924<\/p>\n

[2012] Karl-Theodor Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, http:\/\/arxiv.org\/abs\/1208.0434\u00a0 “The space X of all metric measure spaces … is proven to have nonnegative curvature in the sense of Alexandrov.” Geodesics are linear interpolations on products.<\/p>\n

[2012] David Bate [student of David Preiss at Warwick], Structure of measures in Lipschitz differentiability spaces. http:\/\/arxiv.org\/abs\/1208.1954<\/p>\n

[2012] Haizhong Li and Yong Wei, f-minimal surface and manifold with positive m-Bakry-\u00c9mery Ricci curvature, http:\/\/arxiv.org\/abs\/1209.0895v1\u00a0 Generalize to density Choi-Schoen theorem that minimal surfaces of fixed topological type in closed 3-manifold smoothly compact.<\/p>\n

[2012] Christian Ketterer [student of Sturm], Ricci curvature bounds for warped products, http:\/\/arxiv.org\/abs\/1209.1325<\/p>\n

[2012] Zahra Sinaei,Harmonic maps on smooth metric measure spaces and their convergence, http:\/\/arxiv.org\/abs\/1209.5893\u00a0 [Good references.]<\/p>\n

[2012] Luigi Ambrosio, Nicola Gigli, Giuseppe Savar\u00e9, Bakry-\u00c9mery curvature-dimension condition and Riemannian Ricci curvature bounds, http:\/\/arxiv.org\/abs\/1209.5786<\/p>\n

[2012] Jose M. Espinar, Weighted stable CMC surfaces in manifolds with density, http:\/\/arxiv.org\/abs\/1209.6162<\/p>\n

[2012] Alessio Figalli, Emanuel Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones http:\/\/arxiv.org\/abs\/1210.3113\u00a0 Degenerate Wulff shape!<\/p>\n

[2012] Xu Cheng, Tito Mejia, Detang Zhou, Stability and compactness for complete f-minimal surfaces, http:\/\/arxiv.org\/abs\/1210.8076. Cites Morgan [2005].<\/p>\n

[2012] Alexandru Krist\u00e1ly and Shin-ichi Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications, http:\/\/arxiv.org\/abs\/1211.3171v1<\/p>\n

[2012] Ovidiu Munteanu and Jiaping Wang, Geometry of manifolds with densities, http:\/\/arxiv.org\/abs\/1211.3996\u00a0 Bochner, Laplacian, entropy; Euclidean vol growth, splitting.<\/p>\n

[2012] Gang Liu, Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor,\u00a0Comm. Anal. Geom.,\u00a0http:\/\/arxiv.org\/abs\/1211.3770. 2nd varn, splitting.<\/p>\n

[2012] Jon Wolfson, Eigenvalue gap theorems for a class of non symmetric elliptic operators on convex domains, http:\/\/arxiv.org\/abs\/1212.1669. “The class of operators includes the Bakry-Emery laplacian.”<\/p>\n

[2013]\u00a0Matheus Vieira,\u00a0Harmonic forms on manifolds with non-negative Bakry\u2013\u00c9mery\u2013Ricci curvature,\u00a0Arch. Math. 101 (2013), 581\u2013590.<\/p>\n

[2013] Qin Huang, Qihua Ruan, Applications of Some Elliptic Equations in Riemannian Manifolds, http:\/\/arxiv.org\/abs\/1301.1117\u00a0 (Heintze-Karcher for m-Ricci, later by Batista-Cavalcante; thank and cite Ma.)<\/p>\n

[2013] Fabio Cavalletti and Martin Huesmann, Existence and uniqueness of optimal transport maps, http:\/\/arxiv.org\/abs\/1301.1782v1<\/p>\n

[2013] Debora Impera and Michele Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, http:\/\/arxiv.org\/abs\/1302.6160<\/p>\n

[2013] Mingliang Cai, On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature, http:\/\/arxiv.org\/abs\/1303.2728<\/p>\n

[2013] Matthias Erbar, Kazumasa Kuwada, Karl-Theodor Sturm, On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner’s Inequality on Metric Measure Spaces, http:\/\/arxiv.org\/abs\/1303.4382\u00a0 Good overview introduction.<\/p>\n

[2013] Giuseppe Savar\u00e9, Self-improvement of the Bakry-\u00c9mery condition and Wasserstein contraction of the heat flow in RCD(K,\\infty) metric measure spaces, http:\/\/arxiv.org\/abs\/1304.0643<\/p>\n

[2013] Antonio Ca\u00f1ete, C\u00e9sar Rosales, Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities, http:\/\/arxiv.org\/abs\/1304.1438 Good references.<\/p>\n

[2013] Xavier Cabre, Xavier Ros-Oton, Joaquim Serra, Sharp isoperimetric inequalities via the ABP method, http:\/\/arxiv.org\/abs\/1304.1724\u00a0 Uses Figalli-Indrei degenerate Wulff shape to handle nonsmooth cones.<\/p>\n

[2013] Marcio Batista, Heudson Mirandola [Brazil], A Sobolev-type inequality for submanifolds in weighted Riemannian Manifolds, http:\/\/arxiv.org\/abs\/1304.2271<\/p>\n

[2013] Jia-Yong Wu, L^p-Liouville theorems on complete smooth metric measure spaces, http:\/\/arxiv.org\/abs\/1305.0616<\/p>\n

[2013] Xu Cheng, Tito Mejia, Detang Zhou, Simons’ type equation for f-minimal hypersurfaces and applications, http:\/\/arxiv.org\/abs\/1305.2379v1<\/p>\n

[2013] Xu Cheng, Detang Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces, http:\/\/arxiv.org\/abs\/1305.4116<\/p>\n

[2013] Nicola Garofalo, Andrea Mondino, Li-Yau and Harnack type inequalities in RCD^*(K,N) metric measure spaces, http:\/\/arxiv.org\/abs\/1306.0494<\/p>\n

[2013] Haizhong Li, Yong Wei, Sharp diameter estimates for compact manifold with boundary, http:\/\/arxiv.org\/abs\/1306.3715\u00a0 Ruan generalized Cheng to m-Ricci; LW generalize to manifolds with boundary.<\/p>\n

[2013] Ye-Lin Ou, On f-biharmonic maps and f-biharmonic submanifolds, http:\/\/arxiv.org\/abs\/1306.3549, after Wei-Jun Lu, On f-Biharmonic maps between Riemannian manifolds , arXiv:1305.5478<\/p>\n

[2013] Marcio Batista, Marcos P. Cavalcante, The Heintze-Karcher-Ros Inequality and Eigenvalue Estimates in Weighted Manifolds http:\/\/arxiv.org\/abs\/1306.4874 (Heintze-Karcher for m-Ricci, earlier by Huang Ruan.)<\/p>\n

[2013]\u00a0Aaron Naber,\u00a0Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces,\u00a0http:\/\/arxiv.org\/abs\/1306.6512<\/p>\n

[2013] Bingyu Song, Guofang Wei, Guoqiang Wu, Monotonicity Formulas for Bakry-Emery Ricci Curvature, http:\/\/arxiv.org\/abs\/1307.0477<\/p>\n

[2013] Xu Cheng, Detang Zhou, Stability properties and gap theorem for complete f-minimal hypersurfaces, http:\/\/arxiv.org\/abs\/1307.5099v1<\/p>\n

[2013]\u00a0Matthew McGonagle, John Ross,\u00a0The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space,\u00a0http:\/\/arxiv.org\/abs\/1307.7088<\/p>\n

[2013] Bobo Hua, Martin Kell, Chao Xia, Harmonic functions on metric measure spaces, http:\/\/arxiv.org\/abs\/1308.3607<\/p>\n

[2013]\u00a0Emanuel Milman and Liran Rotem,\u00a0Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures,\u00a0http:\/\/arxiv.org\/abs\/1308.5695. Also negative dimension.<\/p>\n

[2013] Luigi Ambrosio, Andrea Mondino, Giuseppe Savar\u00e9, On the Bakry-\u00c9mery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces, http:\/\/arxiv.org\/abs\/1309.4664<\/p>\n

[2013] Maree Jaramillo, Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below, http:\/\/arxiv.org\/abs\/1309.6685<\/p>\n

[2013] Shouhei Honda, Cheeger constant, p<\/em>-Laplacian, and Gromov-Hausdorff convergence, http:\/\/arxiv.org\/abs\/1310.0304<\/p>\n

[2013]\u00a0Bruno Colbois, Ahmad El Soufi, Alessandro Savo,\u00a0Eigenvalues of the Laplacian on a compact manifold with density, http:\/\/arxiv.org\/abs\/1310.1490<\/p>\n

[2013]\u00a0Alexander V. Kolesnikov, Emanuel Milman,\u00a0Poincar\u00e9 and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary,\u00a0http:\/\/arxiv.org\/abs\/1310.2526. Unified positive and negative dimension.<\/p>\n

[2013]\u00a0Fabio Cavalletti,\u00a0Monge problem in metric measure spaces with Riemannian curvature-dimension condition,\u00a0http:\/\/arxiv.org\/abs\/1310.4036<\/p>\n

[2013]\u00a0Shin-ichi Ohta,\u00a0(K<\/em>,N<\/em>)-convexity and the curvature-dimension condition for negative N<\/em>,\u00a0http:\/\/arxiv.org\/abs\/1310.7993<\/p>\n

[2013]\u00a0\u00a0William Wylie,\u00a0Sectional curvature for Riemannian manifolds with density,\u00a0http:\/\/arxiv.org\/abs\/1311.0267<\/a><\/p>\n

[2013]\u00a0Christian Ketterer,\u00a0Cones over metric measure spaces and the maximal diameter theorem,\u00a0http:\/\/arxiv.org\/abs\/1311.1307<\/p>\n

[2013] Katherine Castro and C\u00e9sar Rosales,\u00a0Free boundary stable hypersurfaces in manifolds with density and rigidity results,\u00a0http:\/\/arxiv.org\/abs\/1311.1952<\/p>\n

[2013]\u00a0Gregory R. Chambers,\u00a0Proof of the Log-Convex Density Conjecture,\u00a0https:\/\/arxiv.org\/abs\/1311.4012,\u00a0J. Eur. Math. Soc. 21 (2019), 2301\u20132332.<\/p>\n

[2013]\u00a0M Rupert and E Woolgar,\u00a0Bakry-\u00c9mery black holes,\u00a0http:\/\/arxiv.org\/abs\/1310.3894 [2013]\u00a0E Woolgar,\u00a0Scalar-tensor gravitation and the Bakry-Emery-Ricci tensor,\u00a0http:\/\/arxiv.org\/abs\/1302.1893 [2013]\u00a0Nicola Gigli, Andrea Mondino, Giuseppe Savar\u00e9,\u00a0Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows,\u00a0http:\/\/arxiv.org\/abs\/1311.4907<\/p>\n

[2013]\u00a0Martin Kell,\u00a0On Interpolation and Curvature via Wasserstein Geodesics,\u00a0http:\/\/arxiv.org\/abs\/1311.5407 “The proof of the Borell\u2013Brascamp\u2013Lieb (BBL) inequality for Riemannian manifolds by Cordero-Erausquin-McCann-Schmuckenschl\u00e4ger [CEMS01], and later for Finsler manifolds by Ohta [Oht09], led Lott-Villani [LV09, LV07] and Sturm [Stu06a, Stu06b] to a new notion of a lower bound on the generalized Ricci curvature for metric measure spaces, called curvature dimension.”<\/p>\n

[2013]\u00a0Katrin F\u00e4ssler, Pekka Koskela, Enrico Le Donne.\u00a0Nonexistence of quasiconformal maps between certain metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1312.1305v1<\/a><\/p>\n

[2013]\u00a0Gregory J Galloway and Eric Woolgar,\u00a0Cosmological singularities in Bakry-\u00c9mery spacetimes,\u00a0http:\/\/arxiv.org\/abs\/1312.3410<\/p>\n

[2013]\u00a0Guy C. David,\u00a0Bi-Lipschitz Pieces between Manifolds,\u00a0http:\/\/arxiv.org\/abs\/1312.3911v1<\/a><\/p>\n

[2013]\u00a0Jeff Cheeger, Bruce Kleiner,\u00a0Inverse limit spaces satisfying a Poincare inequality,\u00a0http:\/\/arxiv.org\/abs\/1312.5227v1<\/a><\/p>\n

[2013]\u00a0Jia-Yong Wu, Peng Wu,\u00a0Heat Kernels on Smooth Metric Measure Spaces with Nonnegative Curvature,\u00a0http:\/\/arxiv.org\/abs\/1401.6155<\/p>\n

[2014]\u00a0Takashi Shioya,\u00a0Metric measure limits of spheres and complex projective spaces,\u00a0http:\/\/arxiv.org\/abs\/1402.0611 [2014]\u00a0\u00a0Yashar Memarian,\u00a0Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds,\u00a0http:\/\/arxiv.org\/abs\/1402.4947 [apparently incorrect]<\/p>\n

[2014]\u00a0Christian Ketterer, Tapio Rajala,\u00a0Failure of topological rigidity results for the measure contraction property,\u00a0http:\/\/arxiv.org\/abs\/1403.3105<\/p>\n

[2014]\u00a0C\u00e9sar Rosales,\u00a0Isoperimetric and stable sets for log-concave perturbations of Gaussian measures,\u00a0http:\/\/arxiv.org\/abs\/1403.4510<\/p>\n

[2014]\u00a0Ayato Mitsuishi,\u00a0Current and measure homologies,\u00a0http:\/\/arxiv.org\/abs\/1403.5518<\/p>\n

[2014] Karl-Theodor Sturm,\u00a0Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions,\u00a0http:\/\/arxiv.org\/abs\/1405.0459. “There are two canonical ways to define the heat flow on a mms, either as the gradient flow for the energy or as the gradient flow for the entropy in the Wasserstein space. …in great generality both approaches will coincide.”<\/p>\n

[2014]\u00a0Yu Kitabeppu, Sajjad Lakzian,\u00a0Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups,\u00a0http:\/\/arxiv.org\/abs\/1405.0897<\/p>\n

[2014]\u00a0Renjin Jiang,\u00a0The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces,\u00a0http:\/\/arxiv.org\/abs\/1405.0684<\/p>\n

[2014]\u00a0Andrea Mondino, Aaron Naber,\u00a0Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I,\u00a0http:\/\/arxiv.org\/abs\/1405.2222<\/p>\n

[2014] Yi Li,\u00a0Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature,\u00a0http:\/\/arxiv.org\/abs\/1406.0125<\/p>\n

[2014]\u00a0Jeffrey S. Case, Sun-Yung Alice Chang,\u00a0On fractional GJMS operators,\u00a0http:\/\/arxiv.org\/abs\/1406.1846<\/p>\n

[2014]\u00a0Luigi Ambrosio, Andrea Pinamonti, Gareth Speight,\u00a0Weighted Sobolev Spaces on Metric Measure Spaces,\u00a0http:\/\/arxiv.org\/abs\/1406.3000<\/a><\/p>\n

[2014]\u00a0Jia-Yong Wu, Peng Wu,\u00a0Heat kernel on smooth metric measure spaces and applications,\u00a0http:\/\/arxiv.org\/abs\/1406.5801<\/p>\n

[2014]\u00a0Nicola Gigli, Bangxian Han,\u00a0The continuity equation on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1406.6350<\/p>\n

[2014]\u00a0Nicola Gigli,\u00a0Nonsmooth differential geometry – An approach tailored for spaces with Ricci curvature bounded from below,\u00a0http:\/\/arxiv.org\/abs\/1407.0809<\/p>\n

[2014]\u00a0Fr\u00e9d\u00e9ric Bernicot, Thierry Coulhon, Dorothee FreyGradient estimates, Poincar\u00e9 inequalities, De Giorgi property and their consequences,\u00a0http:\/\/arxiv.org\/abs\/1407.3906<\/p>\n

[2014]\u00a0Nicola Gigli, Bangxian Han,\u00a0Independence on p of weak upper gradients on RCD spaces,\u00a0http:\/\/arxiv.org\/abs\/1407.7350<\/p>\n

[2014]\u00a0A. Barros, R. Batista, E. Ribeiro Jr.,\u00a0Bounds on volume growth of geodesic balls for Einstein warped products,\u00a0http:\/\/arxiv.org\/abs\/1408.1621<\/p>\n

[2014]\u00a0Erlend Grong, Anton Thalmaier,\u00a0Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Parts I and II,\u00a0http:\/\/arxiv.org\/abs\/1408.6873,\u00a0http:\/\/arxiv.org\/abs\/1408.6872<\/p>\n

[2014]\u00a0David Bate, Sean Li,\u00a0Characterizations of rectifiable metric spaces,\u00a0http:\/\/arxiv.org\/abs\/1409.4242 [2014]\u00a0Emanuel Milman,\u00a0Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension,\u00a0http:\/\/arxiv.org\/abs\/1409.4109<\/p>\n

[2014]\u00a0Jeffrey S. Case,\u00a0A notion of the weighted \u03c3_k-curvature for manifolds with density,\u00a0http:\/\/arxiv.org\/abs\/1409.4455<\/p>\n

[2014]\u00a0Simone Di Marino,\u00a0Sobolev and BV spaces on metric measure spaces via derivations and integration by parts,\u00a0http:\/\/arxiv.org\/abs\/1409.5620<\/p>\n

[2014]\u00a0Jing Mao,\u00a0The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature,\u00a0http:\/\/arxiv.org\/abs\/1409.5741<\/p>\n

[2014]\u00a0Takashi Shioya,\u00a0Metric measure geometry,\u00a0http:\/\/arxiv.org\/abs\/1410.0428<\/p>\n

[2014]\u00a0Lee Kennard, William Wylie,\u00a0Positive weighted sectional curvature, http:\/\/arxiv.org\/abs\/1410.1558<\/a><\/p>\n

[2014]\u00a0Antonio G. Ache, Micah W. Warren,\u00a0Coarse Ricci curvature and the manifold learning problem,\u00a0http:\/\/arxiv.org\/abs\/1410.3351<\/p>\n

[2014]\u00a0Shouhei Honda,\u00a0Elliptic PDEs on compact Ricci limit spaces and applications,\u00a0http:\/\/arxiv.org\/abs\/1410.3296<\/p>\n

[2014]\u00a0Karl-Theodor Sturm,\u00a0Gradient Flows for Semiconvex Functions on Metric Measure Spaces – Existence, Uniqueness and Lipschitz Continuity,\u00a0http:\/\/arxiv.org\/abs\/1410.3966<\/p>\n

[2014]\u00a0Christian Ketterer,\u00a0Obata’s rigidity theorem for metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1410.5210<\/p>\n

[2014]\u00a0Jia-Yong Wu, Peng Wu,\u00a0On L^p-Liouville property for smooth metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1410.7305<\/p>\n

[2014] Paul W.Y. Lee,\u00a0On measure contraction property without Ricci curvature lower bound,\u00a0http:\/\/arxiv.org\/abs\/1412.4345<\/p>\n

[2014]\u00a0Dominique Bakry (IMT), Fran\u00e7ois Bolley (LPMA), Ivan Gentil (ICJ),\u00a0The Li-Yau inequality and applications under a curvature-dimension condition,\u00a0http:\/\/arxiv.org\/abs\/1412.5165<\/p>\n

[2015]\u00a0Fabio Cavalletti, Andrea Mondino,\u00a0Measure rigidity of Ricci curvature lower bounds,\u00a0http:\/\/arxiv.org\/abs\/1501.03338<\/p>\n

[2015]\u00a0William Wylie,\u00a0Some curvature pinching results for Riemannian manifolds with density,\u00a0http:\/\/arxiv.org\/abs\/1501.06079<\/p>\n

[2015]\u00a0S\u00e9rgio Mendon\u00e7a,\u00a0Splitting, parallel gradient and Bakry-Emery Ricci curvature,\u00a0http:\/\/arxiv.org\/abs\/1502.00185<\/p>\n

[2015] Guy C. David, Lusin-type theorems for Cheeger derivatives on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1502.00694<\/p>\n

[2015]\u00a0Fabio Cavalletti, Andrea Mondino,\u00a0Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds,\u00a0http:\/\/arxiv.org\/abs\/1502.06465.\u00a0Invent. Math. 208 (2017), no. 3, 803\u2013849. Generalize Milman smooth to non-branching CD.<\/p>\n

[2015]\u00a0Yin Jiang, Hui-Chun Zhang,\u00a0Sharp Spectral Gaps on Metric Measure Spaces,\u00a0http:\/\/arxiv.org\/abs\/1503.00203<\/p>\n

[2015]\u00a0Ezequiel Barbosa, Ben Sharp, Yong Wei,\u00a0Smooth compactness of f-minimal hypersurfaces with bounded f-index,\u00a0http:\/\/arxiv.org\/abs\/1503.01945<\/p>\n

[2015]\u00a0M. Carmen Domingo-Juan, Vicente Miquel,\u00a0Reilly’s type inequality for the Laplacian associated to a density related with shrinkers for MCF,\u00a0http:\/\/arxiv.org\/abs\/1503.01332<\/p>\n

[2015]\u00a0Vicente Miquel, Francisco Vi\u00f1ado-Lereu,\u00a0The curve shortening problem associated to a density,\u00a0http:\/\/arxiv.org\/abs\/1503.02429<\/p>\n

[2015] M. P. Cavalcante, J. Q. Oliveira, M. S. Santos, Compactness in weighted manifolds and applications, Results Math. (2015).<\/p>\n

[2015]\u00a0 Wyatt Boyer, Bryan Brown, Gregory R. Chambers, Alyssa Loving, Sarah Tammen,\u00a0Isoperimetric Regions in \u211d^n with density r^p,\u00a0https:\/\/arxiv.org\/abs\/1504.01720,\u00a0Anal. Geom. Metr. Spaces 4 (2016), 236\u2013265.<\/p>\n

[2015]\u00a0Homare Tadano,\u00a0Remark on a diameter bound for complete manifolds with positive Bakry-\u00c9mery Ricci curvature,\u00a0http:\/\/arxiv.org\/abs\/1504.05384<\/p>\n

[2015]\u00a0Fabio Cavalletti, Andrea Mondino,\u00a0Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds,\u00a0http:\/\/arxiv.org\/abs\/1505.02061<\/p>\n

[2015]\u00a0Emanuel Milman,\u00a0Harmonic Measures on the Sphere via Curvature-Dimension, http:\/\/arxiv.org\/abs\/1505.04335v1<\/a><\/p>\n

[2015] Christian Ketterer,\u00a0On the geometry of metric measure spaces with variable curvature bounds,\u00a0http:\/\/arxiv.org\/abs\/1506.03279<\/p>\n

[2015] William Wylie,\u00a0A warped product version of the Cheeger-Gromoll splitting theorem,\u00a0http:\/\/arxiv.org\/abs\/1506.03800<\/p>\n

[2015] Luigi Ambrosio, Matthias Erbar, Giuseppe Savar\u00e9,\u00a0Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces,\u00a0http:\/\/arxiv.org\/abs\/1506.05932<\/p>\n

[2015]\u00a0Emanuel Milman,\u00a0Spectral Estimates, Contractions and Hypercontractivity,\u00a0https:\/\/arxiv.org\/abs\/1508.00606<\/p>\n

[2015] Christian Ketterer,\u00a0Evolution variational inequality and Wasserstein control in variable curvature context,\u00a0http:\/\/arxiv.org\/abs\/1509.02178<\/span><\/p>\n

[2015] Luigi Ambrosio, Andrea Mondino, Giuseppe Savar\u00e9,\u00a0Nonlinear diffusion equations and curvature conditions in metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1509.07273<\/span><\/p>\n

[2015] Max Fathi, Yan Shu,\u00a0Curvature and transport inequalities for Markov chains in discrete spaces,\u00a0http:\/\/arxiv.org\/abs\/1509.07160<\/span><\/p>\n

[2015] Shouhei Honda,\u00a0Spectral convergence under bounded Ricci curvature,\u00a0http:\/\/arxiv.org\/abs\/1510.05349<\/span><\/p>\n

[2015] Bang-Xian Han, Conformal transformation on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1511.03115<\/p>\n

[2015] Feng Du, Jing Mao, Qiaoling Wang, Chuanxi Wu, The Gagliardo-Nirenberg inequality on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1511.04696<\/p>\n

[2015] Martin Kell, On Cheeger and Sobolev differentials in metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1512.00828<\/p>\n

[2015] Yuzhao Wang, Huaiqian Li, Lower Bound Estimates for The First Eigenvalue of The Weighted p-Laplacian on Smooth Metric Measure Space,\u00a0http:\/\/arxiv.org\/abs\/1512.01031 [2015]\u00a0Nicola Gigli, Guido de Philippis,\u00a0From volume cone to metric cone in the non smooth setting,\u00a0http:\/\/arxiv.org\/abs\/1512.03113<\/p>\n

[2015] Renjin Jiang, Huichun Zhang, Hamilton’s Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces,\u00a0http:\/\/arxiv.org\/abs\/1512.08306<\/p>\n

[2016] Tomasz Adamowicz, Micha\u0142 Gaczkowski, Przemys\u0142aw G\u00f3rka, Harmonic functions on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1601.03919<\/p>\n

[2016]\u00a0Shin-ichi Ohta, A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetric inequality [for Finsler manifolds with density],\u00a0http:\/\/arxiv.org\/abs\/1602.00390<\/p>\n

[2016] Hui-Chun Zhang, Xi-Ping Zhu, Local<\/em> Li-Yau’s estimates on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1602.05347<\/p>\n

[2016] William Wylie, Dmytro Yeroshkin, On the geometry of Riemannian manifolds with density, dedicated to Frank Morgan,\u00a0http:\/\/arxiv.org\/abs\/1602.08000<\/p>\n

[2016] Qintao Deng, Fernando Galaz-Garcia, Luis Guijarro, Michael Munn, Three-Dimensional Alexandrov spaces with positive or nonnegative Ricci curvature,\u00a0http:\/\/arxiv.org\/abs\/1602.07724v1.\u00a0“We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the \u2217(K,N) sense.”<\/p>\n

[2016] Erbar Matthias, Juillet Nicolas, Smoothing and non-smoothing via a flow tangent to the Ricci flow,\u00a0http:\/\/arxiv.org\/abs\/1603.00280<\/p>\n

[2016] Karl-Theodor Sturm, Super-Ricci Flows for Metric Measure Spaces. I,\u00a0http:\/\/arxiv.org\/abs\/1603.02193 [2016] Jeffrey S. Case, A weighted renormalized curvature for manifolds with density,\u00a0http:\/\/arxiv.org\/abs\/1603.02989<\/p>\n

[2016] Yu Kitabeppu, A Bishop type inequality on metric measure spaces with Ricci curvature bounded below,\u00a0http:\/\/arxiv.org\/abs\/1603.04162<\/p>\n

[2016]\u00a0Xian-tao Huang,\u00a0Noncompact RCD(0,N) spaces with linear volume growth,\u00a0http:\/\/arxiv.org\/abs\/1603.05221<\/p>\n

[2016]\u00a0Luigi Ambrosio, Federico Stra, Dario Trevisan,\u00a0Weak and strong convergence of derivations and stability of flows with respect to MGH convergence,\u00a0http:\/\/arxiv.org\/abs\/1603.05561<\/p>\n

[2016]\u00a0Luigi Ambrosio, Nicola Gigli, Simone Di Marino,\u00a0Perimeter as relaxed Minkowski content in metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1603.08412 [2016]\u00a0Fabrice Baudoin, Daniel J. Kelleher,\u00a0Poincar\u00e9 Duality, Bakry-\u00c9mery Estimates and Isoperimetry on Fractals,\u00a0http:\/\/arxiv.org\/abs\/1604.02520<\/p>\n

[2016]\u00a0Ronen Eldan, James R. Lee, Joseph Lehec,\u00a0Transport-entropy inequalities and curvature in discrete-space Markov chains,\u00a0http:\/\/arxiv.org\/abs\/1604.06859<\/p>\n

[2016]\u00a0Andrea Mondino, Guofang Wei,\u00a0On the universal cover and the fundamental group of an RCD\u2217(K,N)-space,\u00a0http:\/\/arxiv.org\/abs\/1605.02854 \u00a0Nice intro.<\/p>\n

[2016]\u00a0Luigi Ambrosio, Andrea Mondino,\u00a0Gaussian-type Isoperimetric Inequalities in RCD(K,\u221e) probability spaces for positive K,\u00a0http:\/\/arxiv.org\/abs\/1605.02852 \u00a0After Bakry-\u00c9mery.<\/p>\n

[2016]\u00a0Sajjad Lakzian, Zachary Mcguirk,\u00a0Global Poincar\u00e9 inequality on Graphs via Conical Curvature-Dimension Conditions,\u00a0http:\/\/arxiv.org\/abs\/1605.05432<\/p>\n

[2016]\u00a0Luigi Ambrosio, Shouhei Honda,\u00a0New stability results for sequences of metric measure spaces with uniform Ricci bounds from below,\u00a0http:\/\/arxiv.org\/abs\/1605.07908<\/p>\n

[2016]\u00a0Max Grieshammer, Thomas Rippl,\u00a0Partial orders on metric measure spaces,\u00a0http:\/\/arxiv.org\/abs\/1605.08989<\/p>\n

[2016]\u00a0Qi S Zhang, Meng Zhu,\u00a0New Volume Comparison results and Applications to degeneration of Riemannian metrics,\u00a0http:\/\/arxiv.org\/abs\/1605.09420<\/p>\n

[2016]\u00a0A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro,\u00a0Some isoperimetric inequalities on \u211dN with respect to weights |x|^\u03b1,\u00a0https:\/\/arxiv.org\/abs\/1606.02195v2<\/p>\n

[2016]\u00a0Lingzhong Zeng,\u00a0The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces,\u00a0http:\/\/arxiv.org\/abs\/1606.06429<\/p>\n

[2016]\u00a0Martin Kell, Andrea Mondino,\u00a0On the volume measure of non-smooth spaces with Ricci curvature bounded below,\u00a0http:\/\/arxiv.org\/abs\/1607.02036<\/p>\n

[2016]\u00a0Nicola Gigli, Enrico Pasqualetto,\u00a0Behaviour of the reference measure on spaces under charts,\u00a0http:\/\/arxiv.org\/abs\/1607.05188<\/p>\n

[2016]\u00a0Sylvester Eriksson-Bique,\u00a0Classifying Poincar\u00e9 Inequalities and the local geometry of RNP-Differentiability Spaces,\u00a0http:\/\/arxiv.org\/abs\/1607.07428 [2016]\u00a0Vicente Miquel, Francisco Vi\u00f1ado-Lereu,\u00a0Type I singularities in the curve shortening flow associated to a density,\u00a0http:\/\/arxiv.org\/abs\/1607.08402<\/p>\n

[2016]\u00a0Robert Haslhofer, Aaron Naber,\u00a0Ricci Curvature and Bochner Formulas for Martingales,\u00a0http:\/\/arxiv.org\/abs\/1608.04371<\/p>\n

[2016]\u00a0Luis Guijarro, Jaime Santos-Rodr\u00edguez,\u00a0On the isometry group of RCD\u2217(K,N)-spaces,\u00a0http:\/\/arxiv.org\/abs\/1608.06467<\/p>\n

[2016]\u00a0Shiping Liu, Florentin M\u00fcnch, Norbert Peyerimhoff,\u00a0Bakry-Emery curvature and diameter bounds on graphs,\u00a0http:\/\/arxiv.org\/abs\/1608.07778<\/p>\n

[2016]\u00a0Fabio Cavalletti, Andrea Mondino,\u00a0Optimal maps in essentially non-branching spaces,\u00a0http:\/\/arxiv.org\/abs\/1609.00782<\/p>\n

[2016]\u00a0Gerardo Sosa<\/a>,\u00a0The isometry group of an RCD*-space is Lie,\u00a0http:\/\/arxiv.org\/abs\/1609.02098<\/p>\n

[2016]\u00a0Luk\u00e1\u0161 Mal\u00fd, Nageswari Shanmugalingam,\u00a0Neumann problem for p<\/em>-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity,\u00a0http:\/\/arxiv.org\/abs\/1609.06808<\/p>\n

[2016] Li Ma,\u00a0Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers,\u00a0https:\/\/arxiv.org\/abs\/1609.09332<\/p>\n

[2016]\u00a0Shouhei Honda,\u00a0Ricci curvature and Orientability,\u00a0https:\/\/arxiv.org\/abs\/1610.02932<\/p>\n

[2016]\u00a0Christian Ketterer, Andrea Mondino,\u00a0Sectional and intermediate Ricci curvature lower bounds via Optimal Transport,\u00a0https:\/\/arxiv.org\/abs\/1610.03339<\/p>\n

[2016]\u00a0Nguyen Thac Dung, Nguyen Ngoc Khanh, Quoc Anh Ng\u00f4,\u00a0Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1610.03199<\/p>\n

[2016]\u00a0Nguyen Thac Dung, Kieu Thi Thuy Linh, Ninh Van Thu,\u00a0Gradient estimates for some evolution equations on complete smooth metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1610.03198<\/p>\n

[2016]\u00a0Jia-Yong Wu,\u00a0Comparison geometry for integral Bakry-\u00c9mery Ricci tensor bounds,\u00a0https:\/\/arxiv.org\/abs\/1610.03926<\/p>\n

[2016]\u00a0Jia-Yong Wu,\u00a0Liouville property for f-harmonic functions with polynomial growth,\u00a0https:\/\/arxiv.org\/abs\/1610.03923<\/p>\n

[2016]\u00a0Fabio Cavalletti, Andrea Mondino,\u00a0Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds,\u00a0https:\/\/arxiv.org\/abs\/1610.05044. For perimeter, not just Minkowski content.<\/p>\n

[2016]\u00a0Guy C. David, Bruce Kleiner,\u00a0Rectifiability of planes and Alberti representations,\u00a0https:\/\/arxiv.org\/abs\/1611.05284<\/p>\n

[2016]\u00a0Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu,\u00a0Balls Isoperimetric in \u211d^n with Volume and Perimeter Densities r^m and r^p,\u00a0https:\/\/arxiv.org\/abs\/1610.05830<\/p>\n

[2016]\u00a0Ushio Tanaka,\u00a0Gromov’s Problem: Bound the Expansion Coefficient from below in terms of the Observable Diameter of a Metric Measure Space, and its Diameter Bounds,\u00a0https:\/\/arxiv.org\/abs\/1611.05596<\/p>\n

[2016]\u00a0Panu Lahti,\u00a0Strong approximation of sets of finite perimeter in metric spaces,\u00a0https:\/\/arxiv.org\/abs\/1611.06162 [2016]\u00a0Nicola Gigli, Enrico Pasqualetto,\u00a0Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1611.09645<\/p>\n

[2016]\u00a0Panu Lahti,\u00a0Strict and pointwise convergence of BV functions in metric spaces,\u00a0https:\/\/arxiv.org\/abs\/1612.06447<\/p>\n

[2016]\u00a0Fabio Cavalletti, Emanuel Milman,\u00a0The Globalization Theorem for the Curvature Dimension Condition,\u00a0https:\/\/arxiv.org\/abs\/1612.07623<\/p>\n

[2017]\u00a0Hil\u00e1rio Alencar, Adina Rocha,\u00a0The f-Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons,\u00a0https:\/\/arxiv.org\/abs\/1701.00373<\/p>\n

[2017]\u00a0Hui-Chun Zhang, Xi-Ping Zhu,\u00a0Weyl’s law on RCD\u2217(K,N) metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1701.01967<\/p>\n

[2017]\u00a0Jia-Cheng Huang, Hui-Chun Zhang,\u00a0Sharp gradient estimate for heat kernels on RCD\u2217(K,N) metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1701.01803<\/p>\n

[2017]\u00a0Luigi Ambrosio, Shouhei Honda, David Tewodrose,\u00a0Short-time behavior of the heat kernel and Weyl’s law on RCD\u2217(K,N)-spaces,\u00a0https:\/\/arxiv.org\/abs\/1701.03906<\/p>\n

[2017]\u00a0Bang-Xian Han, Andrea Mondino,\u00a0Angles between curves in metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1701.05000<\/p>\n

[2017]\u00a0Davide Barilari, Luca Rizzi,\u00a0Sharp measure contraction property for generalized H-type Carnot groups,\u00a0https:\/\/arxiv.org\/abs\/1702.04401<\/p>\n

[2017]\u00a0Chang-Yu Guo,\u00a0Harmonic mappings between singular metric spaces,\u00a0https:\/\/arxiv.org\/abs\/1702.05086<\/p>\n

[2017]\u00a0\u00c1ngel Arroyo, Jos\u00e9 G. Llorente,\u00a0A priori H\u00f6lder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1702.07175<\/p>\n

[2017]\u00a0Luigi Ambrosio, Shouhei Honda,\u00a0Local spectral convergence in RCD\u2217(K,N) spaces,\u00a0https:\/\/arxiv.org\/abs\/1703.04939<\/p>\n

[2017]\u00a0Panu Lahti,\u00a0Quasiopen sets, bounded variation and lower semicontinuity in metric spaces,\u00a0https:\/\/arxiv.org\/abs\/1703.04675<\/p>\n

[2017]\u00a0Kohei Suzuki,\u00a0Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions,\u00a0https:\/\/arxiv.org\/abs\/1703.07234<\/p>\n

[2017]\u00a0Manuel Ritor\u00e9, Jes\u00fas Yepes Nicol\u00e1s,\u00a0Brunn-Minkowski inequalities in product metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1704.07717<\/p>\n

[2017]\u00a0Nicola Gigli, Chiara Rigoni,\u00a0Recognizing the flat torus among RCD\u2217(0,N) spaces via the study of the first cohomology group,\u00a0https:\/\/arxiv.org\/abs\/1705.04466<\/p>\n

[2017]\u00a0Hiroki Nakajima,\u00a0The maximum of the 1-measurement of a metric measure space,\u00a0https:\/\/arxiv.org\/abs\/1706.01258<\/p>\n

[2017]\u00a0Nicola Gigli, Chiara Rigoni,\u00a0A note about the strong maximum principle on RCD space,\u00a0https:\/\/arxiv.org\/abs\/1706.01998<\/p>\n

[2017]\u00a0Luigi Ambrosio, Shouhei Honda, Jacobus W. Portegies,\u00a0Continuity of nonlinear eigenvalues in CD(K,\u221e) spaces with respect to measured Gromov-Hausdorff convergence,\u00a0https:\/\/arxiv.org\/abs\/1706.08368<\/p>\n

[2017]\u00a0Jia-Yong Wu,\u00a0Myers’ type theorem with the Bakry-\u00c9mery Ricci tensor,\u00a0https:\/\/arxiv.org\/abs\/1706.07897<\/p>\n

[2017]\u00a0Lee Kennard, William Wylie, Dmytro Yeroshkin,\u00a0The weighted connection and sectional curvature for manifolds with density,\u00a0https:\/\/arxiv.org\/abs\/1707.05376<\/p>\n

[2017]\u00a0Eric Woolgar, William Wylie,\u00a0Curvature-dimension bounds for Lorentzian splitting theorems,\u00a0https:\/\/arxiv.org\/abs\/1707.09058<\/p>\n

[2017]\u00a0Guido De Philippis, Nicola Gigli,\u00a0Non-collapsed spaces with Ricci curvature bounded from below,\u00a0https:\/\/arxiv.org\/abs\/1708.02060<\/p>\n

[2017]\u00a0Yu Kitabeppu,\u00a0A sufficient condition to a regular set of positive measure on RCD spaces,\u00a0https:\/\/arxiv.org\/abs\/1708.04309<\/p>\n

[2017]\u00a0Nicola Gigli, Christian Ketterer, Kazumasa Kuwada, Shin-ichi Ohta,\u00a0Rigidity for the spectral gap on RCD(K,\u221e)-spaces,\u00a0https:\/\/arxiv.org\/abs\/1709.04017<\/p>\n

[2017]\u00a0Kohei Suzuki,\u00a0Convergence of Non-Symmetric Diffusion Processes on RCD spaces,\u00a0https:\/\/arxiv.org\/abs\/1709.09536<\/p>\n

[2017]\u00a0Bang-Xian Han,\u00a0Ricci tensor on smooth metric measure space with boundary,\u00a0https:\/\/arxiv.org\/abs\/1709.10143<\/p>\n

[2017]\u00a0Xian-Tao Huang,\u00a0An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth, https:\/\/arxiv.org\/abs\/1710.05830<\/p>\n

[2017]\u00a0Bang-Xian Han,\u00a0Characterizations of monotonicity of vector fields on metric measure space,\u00a0https:\/\/arxiv.org\/abs\/1710.07953<\/p>\n

[2017]\u00a0Helmer Hoppe, Jun Masamune, Stefan Neukamm,\u00a0H-compactness of elliptic operators on weighted Riemannian Manifolds,\u00a0https:\/\/arxiv.org\/abs\/1710.09352<\/p>\n

[2017]\u00a0Luca Rizzi,\u00a0A counterexample to gluing theorems for MCP metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1711.04499<\/p>\n

[2017]\u00a0Willian Isao Tokura, Levi Adriano, Changyu Xia,\u00a0The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces,\u00a0https:\/\/arxiv.org\/abs\/1711.04836<\/p>\n

[2017]\u00a0Vitali Kapovitch, Christian Ketterer,\u00a0RCD meets CAT,\u00a0https:\/\/arxiv.org\/abs\/1712.02839<\/p>\n

[2017]\u00a0Timo Schultz,\u00a0Existence of optimal transport maps in very strict CD(K,\u221e) -spaces,\u00a0https:\/\/arxiv.org\/abs\/1712.03670<\/p>\n

[2017]\u00a0Luigi Ambrosio, Elia Bru\u00e8, Dario Trevisan,\u00a0Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and RCD(K,\u221e) spaces,\u00a0https:\/\/arxiv.org\/abs\/1712.06315<\/p>\n

[2017]\u00a0Cong Hung Mai,\u00a0Rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds,\u00a0https:\/\/arxiv.org\/abs\/1712.06904<\/p>\n

[2017]\u00a0Raphael Bouyrie,\u00a0Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry–Ledoux isoperimetric comparison Theorem,\u00a0https:\/\/arxiv.org\/abs\/1708.07203<\/p>\n

[2017]\u00a0Matthias Erbar, Karl-Theodor Sturm,\u00a0Rigidity of cones with bounded Ricci curvature,\u00a0https:\/\/arxiv.org\/abs\/1712.08093<\/p>\n

[2018]\u00a0Hiroki Nakajima, Takashi Shioya,\u00a0Isoperimetric rigidity and distributions of 1-Lipschitz functions,\u00a0https:\/\/arxiv.org\/abs\/1801.01302<\/p>\n

[2018]\u00a0Antonio Bueno,\u00a0Translating solitons of the mean curvature flow in the space \u210d2\u00d7\u211d,\u00a0https:\/\/arxiv.org\/abs\/1803.02783<\/p>\n

[2018]\u00a0Elia Bru\u00e8, Daniele Semola,\u00a0Regularity of Lagrangian flows over RCD\u2217(K,N) spaces,\u00a0https:\/\/arxiv.org\/abs\/1803.04387<\/p>\n

[2018]\u00a0Sebastiano Don, Davide Vittone,\u00a0A compactness result for BV functions in metric spaces,\u00a0https:\/\/arxiv.org\/abs\/1803.07545<\/p>\n

[2018]\u00a0Debora Impera, Michele Rimoldi, Alessandro Savo,\u00a0Index and first Betti number of f-minimal hypersurfaces and self-shrinkers,\u00a0https:\/\/arxiv.org\/abs\/1803.08268v1<\/p>\n

[2018] Daisuke Kazukawa,\u00a0A new condition for convergence of energy functionals and stability of lower Ricci curvature bound,\u00a0https:\/\/arxiv.org\/abs\/1804.00407<\/p>\n

[2018]\u00a0Elia Bru\u00e8, Daniele Semola,\u00a0Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows,\u00a0https:\/\/arxiv.org\/abs\/1804.07128<\/p>\n

[2018]\u00a0Shouhei Honda,\u00a0Bakry-\u00c9mery conditions on almost smooth metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1804.07043<\/p>\n

[2018]\u00a0Ilaria Mondello (LAMA), J. Bertrand (IMT), C Ketterer, T. Richard (LAMA),\u00a0Stratified spaces and synthetic Ricci curvature bounds,\u00a0https:\/\/arxiv.org\/abs\/1804.08870<\/p>\n

[2018]\u00a0Antoni Kijowski,\u00a0Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure,\u00a0https:\/\/arxiv.org\/abs\/1804.10005. “We conclude with a remarkable observation that strongly harmonic functions in R^<\/strong>n<\/em> possess the mean value property with respect to infinitely many weight functions obtained from a given weight.”<\/p>\n

[2018]\u00a0Angelo Alvino, Friedemann Brock, Francesco Chiacchio, Anna Mercaldo, Maria Rosaria Posteraro,\u00a0The isoperimetric problem for a class of non-radial weights and applications,\u00a0https:\/\/arxiv.org\/abs\/1805.02518v1<\/p>\n

[2018]\u00a0Jhovanny Mu\u00f1oz Posso,\u00a0A generalization of Sobolev trace inequality and Escobar-Riemann mapping type problem on smooth metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1805.03694 [part of PhD thesis under Fernando Cod\u00e1\u00b4Marques]<\/p>\n

[2018]\u00a0Jeff Cheeger, Wenshuai Jiang, Aaron Naber,\u00a0Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below,\u00a0https:\/\/arxiv.org\/abs\/1805.07988<\/p>\n

[2018]\u00a0Ana Hurtado, Vicente Palmer, C\u00e9sar Rosales,\u00a0Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights,\u00a0https:\/\/arxiv.org\/abs\/1805.10055 [2018]\u00a0Katrin F\u00e4ssler, Tuomas Orponen,\u00a0Metric currents and the Poincar\u00e9 inequality,\u00a0https:\/\/arxiv.org\/abs\/1807.02969<\/p>\n

[2018]\u00a0Danka Lu\u010di\u0107, Enrico Pasqualetto,\u00a0Infinitesimal Hilbertianity of weighted Riemannian manifolds,\u00a0https:\/\/arxiv.org\/abs\/1809.05919<\/p>\n

[2018]\u00a0Chris Connell, Xianzhe Dai, Jes\u00fas N\u00fa\u00f1ez-Zimbr\u00f3n, Raquel Perales, Pablo Su\u00e1rez-Serrato, Guofang Wei,\u00a0Maximal volume entropy rigidity for \u2217(\u2212(N\u22121),N) spaces,\u00a0https:\/\/arxiv.org\/abs\/1809.06909<\/p>\n

[2018]\u00a0Jaime Santos-Rodr\u00edguez,\u00a0Invariant measures and lower Ricci curvature bounds,\u00a0https:\/\/arxiv.org\/abs\/1810.11327<\/p>\n

[2018]\u00a0Fabio Cavalletti, Flavia Santarcangelo,\u00a0Isoperimetric inequality under Measure-Contraction property,\u00a0https:\/\/arxiv.org\/abs\/1810.11289. Generalize Levy-Gromov-Milman-CavallettiMondino to MCP spaces.<\/p>\n

[2018]\u00a0L. M. Chasman, Jeffrey J Langford,\u00a0On Clamped Plates with Log-Convex Density,\u00a0https:\/\/arxiv.org\/abs\/1811.06423<\/p>\n

[2018]\u00a0Panu Lahti,\u00a0A sharp Leibniz rule for BV functions in metric spaces,\u00a0https:\/\/arxiv.org\/abs\/1811.07713v1<\/p>\n

[2018]\u00a0Kazumasa Kuwada, Xiang-Dong Li,\u00a0Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces,\u00a0https:\/\/arxiv.org\/abs\/1811.07228<\/p>\n

[2018] Simone Di Marino, Nicola Gigli, Enrico Pasqualetto, Elefterios Soultanis, Infinitesimal Hilbertianity of locally CAT(\u03ba)-spaces, https:\/\/arxiv.org\/abs\/1812.02086<\/p>\n

[2018]\u00a0LI MA,\u00a0LIOUVILLE THEOREMS, VOLUME GROWTH, AND VOLUME COMPARISON FOR RICCI SHRINKERS, PACIFIC J. MATH. 296 dx.doi.org\/10.2140\/pjm.2018.296.357<\/a><\/p>\n

[2019]\u00a0Vitali Kapovitch, Christian Ketterer,\u00a0Weakly noncollapsed RCD spaces with upper curvature bounds,\u00a0https:\/\/arxiv.org\/abs\/1901.06966<\/p>\n

[2019] Bang-XIan Han,\u00a0Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds,\u00a0https:\/\/arxiv.org\/abs\/1902.00942<\/a><\/p>\n

[2019]\u00a0Nicola Gigli,\u00a0Riemann curvature tensor on spaces and possible applications,\u00a0https:\/\/arxiv.org\/abs\/1902.02282<\/p>\n

[2019]\u00a0Friedemann Brock<\/a>, Francesco Chiacchio<\/a>,\u00a0Some weighted isoperimetric problems on \u211d^N+ with stable half balls have no solutions,\u00a0https:\/\/arxiv.org\/abs\/1903.04922<\/p>\n

[2019]\u00a0Adriano Cavalcante Bezerra, Changyu Xia,\u00a0Sharp Estimates for the First Eigenvalues of the Bi-drifting [squared] Laplacian,\u00a0https:\/\/arxiv.org\/abs\/1903.06728<\/p>\n

[2019]\u00a0Giulia Luise, Giuseppe Savar\u00e9,\u00a0Contraction and regularizing properties of heat flows in metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1904.09825<\/p>\n

[2019]\u00a0Shin-ichi Ohta, Asuka Takatsu,\u00a0Equality in the logarithmic Sobolev inequality,\u00a0https:\/\/arxiv.org\/abs\/1904.09400<\/p>\n

[2019]\u00a0Shouhei Honda,\u00a0New differential operator and non-collapsed RCD spaces,\u00a0https:\/\/arxiv.org\/abs\/1905.00123<\/p>\n

[2019]\u00a0Timo Schultz,\u00a0Equivalent definitions of very strict CD(K,N) -spaces,\u00a0https:\/\/arxiv.org\/abs\/1906.07693<\/p>\n

[2019]\u00a0Davide Barilari, Luca Rizzi,\u00a0Bakry-\u00c9mery curvature and model spaces in sub-Riemannian geometry,\u00a0https:\/\/arxiv.org\/abs\/1906.08307<\/p>\n

[2019]\u00a0Mathias Braun, Karen Habermann, Karl-Theodor Sturm,\u00a0Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds,\u00a0https:\/\/arxiv.org\/abs\/1906.09186<\/p>\n

[2019]\u00a0Gioacchino Antonelli, Elia Bru\u00e8, Daniele Semola,\u00a0Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1907.02735<\/p>\n

[2019]\u00a0Vitali Kapovitch, Andrea Mondino,\u00a0On the topology and the boundary of N-dimensional RCD(K,N) spaces,\u00a0https:\/\/arxiv.org\/abs\/1907.02614<\/p>\n

[2019]\u00a0Shouhei Honda, Ilaria Mondello,\u00a0Sphere theorems for RCD spaces,\u00a0https:\/\/arxiv.org\/abs\/1907.03482<\/p>\n

[2019]\u00a0Angelo Alvino, Friedemann Brock, Francesco Chiacchio, Anna Mercaldo, Maria Rosaria Posteraro,\u00a0Some isoperimetric inequalities with respect to monomial weights,\u00a0https:\/\/arxiv.org\/abs\/1907.03659<\/p>\n

[2019]\u00a0Bang-Xian Han, Karl-Theodor Sturm,\u00a0Curvature-dimension conditions for diffusions under time change,\u00a0https:\/\/arxiv.org\/abs\/1907.05761<\/p>\n

[2019]\u00a0Ana Hurtado, Vicente Palmer, C\u00e9sar Rosales,\u00a0Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds,\u00a0https:\/\/arxiv.org\/abs\/1907.07920<\/p>\n

[2019]\u00a0Batu G\u00fcneysu, Max von Renesse,\u00a0Molecules as metric measure spaces with Kato-bounded Ricci curvature,\u00a0https:\/\/arxiv.org\/abs\/1907.09566<\/p>\n

[2019]\u00a0Christian Ketterer,\u00a0The Heintze-Karcher inequality for metric measure spaces,\u00a0https:\/\/arxiv.org\/abs\/1908.06146<\/p>\n

[2019]\u00a0Vitali Kapovitch, Martin Kell, Christian Ketterer,\u00a0On the structure of RCD spaces with upper curvature bounds,\u00a0https:\/\/arxiv.org\/abs\/1908.07036<\/p>\n

[2019] Abdolhakim Shouman, Generalization of Philippin\u2019s results for the first Robin eigenvalue and estimates for eigenvalues of the bi\u2011drifting Laplacian, Annals of Global Analysis and Geometry (2019) 55:805\u2013817 https:\/\/doi.org\/10.1007\/s10455-019-09652-1 1 3<\/p>\n

[2020] Adson Meira \u00b7 Rosivaldo Antonio Gon\u00e7alves, On the space of f-minimal surfaces with bounded f-index in weighted smooth metric spaces, manuscripta math. 162, 559\u2013563 (2020)<\/p>\n

[2020] Kazuhiro Kuwae, Yohei Sakurai, Comparison geometry of manifolds with boundary under lower N-weighted Ricci curvature bounds with \u03b5-range, https:\/\/arxiv.org\/abs\/2011.03730v1<\/p>\n

[2021] Florian Johne, Sobolev inequalities on manifolds with nonnegative Bakry-\u00c9mery Ricci curvature, https:\/\/arxiv.org\/abs\/2103.08496<\/p>\n

[2021] Homare Tadanoa,\u00a0Some compactness theorems via m-Bakry\u2013\u00c9mery and m-modified Ricci curvatures with negative m,\u00a0Differential Geometry and its Applications 75 (2021), 101-720<\/p>\n

[2021] Allan G. Freitasa, Henrique F. de Lima, Eraldo A. Lima Jr., M\u00e1rcio S. Santosa,\u00a0Submanifolds immersed in Riemannian spaces endowed with a Killing vector field: Nonexistence and rigidity,\u00a0Differential Geometry and its Applications 75 (2021) 101-714<\/p>\n

[2021] Fabio Cavalletti, Davide Manini, Isoperimetric inequality in noncompact MCP spaces, https:\/\/arxiv.org\/abs\/2110.07528v1<\/p>\n

[2022] Davide Manini,\u00a0Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature
\nhttps:\/\/arxiv.org\/abs\/2212.05130v1<\/p>\n

[2022] Bang-Xian Han, Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature, https:\/\/arxiv.org\/abs\/2212.11570<\/p>\n

ADDITIONAL PUBLICATIONS IN THE HISTORY OF MANIFOLDS WITH DENSITY<\/strong><\/p>\n

[B3] F. Barthe, Extremal properties of central-half-spaces for product measures, J. Funct. Anal. 182 (2001), 81-107.<\/p>\n

[B4] F. Barthe, Log-concave and spherical models in isoperimetry, Geom. Funct. Anal. 12 (2002), 32-55.<\/p>\n

[B5] F. Barthe and B. Maurey, Some remarks on isoperimetry of gaussian type, Ann. Inst. H. Poincar\u00e9 Prob. Stat. 36, (2000), 419-434.<\/p>\n

[B7] Sergei Bobkov, An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space, Ann Probab. 24 (1997), 206-214. analytic form<\/p>\n

[B8] Sergei Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Prob. 24, 1996, 35-48.<\/p>\n

[B9] Sergei Bobkov, Isoperimetric and analytic inequalities for log-concave probability measures, Ann. Prob. 27, (1999), 1903-1921.<\/p>\n

[B10] Sergei Bobkov and Christian Houdr\u00e9, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129, 1997.<\/p>\n

[CK] E. A. Carlen and C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. 13 (2001), 1-18.<\/p>\n

[D] E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. d\u2019Analyse Math.\u00a058 (1992) 99-119.<\/p>\n

[E] A. Ehrhard, In\u00e9galit\u00e9s isop\u00e9rimetriques et int\u00e9grales de Dirichlet gaussienes, Ann. Sci. Ecole Norm. Sup. 17 (1984), 317-332.<\/p>\n

[FOT] M. Fukushima, Y. Oshima, M. Takeda, \u201cDirichlet forms and symmetric Markov processes\u201d, Studies in Mathematics 19, De Gruyter, 1994.<\/p>\n

[G] Gross, Leonard, Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992), 54–88, Lecture Notes in Math., 1563, Springer, Berlin, 1993. Section 6: “The Gaussian logarithmic Sobolev inequality can be derived in a number of different ways. Its equivalent form, Cor 4.3, which is E. Nelson’s hypercontractivity bound can also be derived in several essentially distinct ways.”<\/p>\n

[LS] Landau, H. J.; Shepp, L. A. On the supremum of a Gaussian process. Sankhya Ser. A 32 1970 369-378. A 0-1 law lemma cited by MR of Borell’s [1975] proof of Gaussian isop. inequality: in Gauss space G^n, if convex C about origin has same volume as halfspace containing origin, then for a > 1, |aC| >= |aH|. Relatively trivial proof just uses Brunn-Minkowski for S^n-1. Result also easily from Gaussian isop. inequality because halfspace minimizes not only perimeter but also x dot unit normal at closest boundary approach to origin and then goes farther in lower density and turns less than C.<\/p>\n

[L0] Michel Ledoux, Isop\u00e9rim\u00e9trie et in\u00e9galit\u00e9s de Sobolev logarithmiques gaussiennes, C. F. Acad. Sci. Paris 206 (1988), 79-82. Houdr\u00e9 tells me this is first observation that some log-Sobolev follows from isoperimetry, though not Gross version.<\/p>\n

[L1] Michel Ledoux, The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, 89, Amer. Math. Soc., 2001.<\/p>\n

[L2] M. Ledoux. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse\u00a0Math. (6), 9(2):305–366, 2000.<\/p>\n

[L3] M. Ledoux. A simple analytic proof of an inequality by P. Buser. Proc. Amer.\u00a0Math. Soc., 121(3):951-959, 1994. First observed that a Gaussian log-Sobolev inequality follows from Gaussian isoperimetric inequality. According to [L2, p. 126], Beckner noticed the easy derivation by plugging f=g^2 into Bobkov (see also [Ros, Thm 3.11]).<\/p>\n

[Lo] J. Lott, Some geometric properties of the Bakry-\u00c9mery-Ricci tensor, Comm. Math. Helv. 78, 2003, 865-883.<\/p>\n

[McK] H. P. McKean, Geometry of differential space, Ann. Prob. 1 (1973), 197-206.<\/p>\n

[N] E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973), 211-227.<\/p>\n

[O] Shin-ichi Ohta,Optimal transport and Ricci curvature in Finsler geometry, Advanced Studies in Pure Mathematics\u00a0 (2010), 1\u201320.<\/p>\n

[Q1] Qian, Zhongmin, On conservation of probability and the Feller property. Ann. Probab. 24 (1996), no. 1, 280-292. “Motivated by the classical Lichnerowicz-Bochner-Weitzenb\u00f6ck formula, Bakry and Emery [3] introduced a bilinear map Gamma_2, the “curvature” operator of the diffusion operator L, taking the place of the Ricci curvature (which corresponds to the case L = Delta). …” (Intro., pp. 281-282]).<\/p>\n

[Q2] Zhongmin Qian, Estimates for weighted volumes and applications, Quart. J. Math. 48 (1997), 235-242.<\/p>\n

[Stam] Stam, A. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2 (1959), 101\u2013112. According to [V, Ch 51, Bib. Notes], “At the end of the fifties, Stam established an inequality which can be recast (after simple changes of functions) as the usual logarithmic Sobolev inequality, found fifteen years later by Gross [1975].<\/p>\n

[Ros] Antonio Ros, The isoperimetric problem, David Hoffman, editor, Global Theory of Minimal Surfaces (Proc. Clay Math. Inst. 2001 Summer School, MSRI), Amer. Math. Soc., 2005, 175-209.<\/p>\n

[RBCM] C\u00e9sar Rosales, Vincent Bayle, Antonio Ca\u00f1ete, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Varn. PDE (2007).<\/p>\n

[V] Cedric Villani, Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. ISBN: 978-3-540-71049-3<\/p>\n","protected":false},"excerpt":{"rendered":"

SELECTED PUBLICATIONS IN THE HISTORY OF MANIFOLDS WITH DENSITY: [1959] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Indo. Control 2 (1959), 101-112, Eqn. 2.3. Gives a version of Gaussian log-Sobolev inequality, used by Perelman, often attributed to Gross [1975] or sometimes Federbush [1969]. [1966] E. Nelson, A […]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-1637","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1637","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=1637"}],"version-history":[{"count":251,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1637\/revisions"}],"predecessor-version":[{"id":3344,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1637\/revisions\/3344"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=1637"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=1637"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=1637"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}