Manifolds with Density: Fuller References


[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 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.

[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, §1.4, p. 182].

[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].

[1970] André Lichnerowicz, Variétés riemanniennes a tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A650-A653. Studies Ric – Hess log density to prove splitting theorems.

[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.

[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.

[1975] Gross, Leonard, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083.  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,  §3.4] traces back to Ehrhard [E, 1984] and Bobkov [B7, 1997], first observed by Ledoux ’94 and Beckner ’96 (published [1999]) (see Morgan Blog and email from Milman). Precursors are Stam [1959] and Federbush [1969] (see Gross survey [G]).

[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  with density) called “homogeneous spaces.”

[1981]  M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser 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.

[1982] Edward Witten. Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661–692 (1983). Pointed out to me by Doan.  d_t = e^-ht d e^ht   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.

[1984] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., 59 (1984) 335-395.

[1985] D. Bakry and M. Émery, Diffusions Hypercontractive, Séminaire de Probabilités XIX 1983/4, Lecture Notes Math. 1123, Springer, 1985, 177-206. Generalized Ricci curvatures. They mention the earlier classical Lichnerowicz-Bochner-Weitzenböck formula right after their statement of their Proposition 3, as pointed out to me by Houdré.

[1989] Kenji Fukaya. Collapsing Riemannian manifolds to ones with lower dimension. I, II. J. Math. Soc. Japan 41 (1989), 333–356. Preceding Cheeger and Colding, 1997.

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

[1991] Isaac Chavel and Edgar A. Feldman, Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke 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.

[1996] D. Bakry and M. Ledoux, Lévy-Gromov isoperimetric inequality for an infinite-dimensional diffusion generator, Inv. Math. 123 (1996), 259-281.

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

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

[2000] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214:3, (2000), 547–563. 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).

[2003] Misha Gromov, Isoperimetry of waists and concentration of maps, Geom. Func. Anal. 13 (2003), 178-215.

[2003] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, (2003). Conceptual development via manifolds with density; specific use of Gross’s Gaussian log-Sobolev inequality. Sect 3.3:  “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].”

[2004] Vincent Bayle, Propriétés de concavité du profil isopérimétrique et applicationes, graduate thesis, Institut Fourier, Universite Joseph-Fourier, Grenoble I, 2004. Beautiful survey plus second variation version of Bakry-Emery and more.

[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].

[2006] Alexander Grigor’yan, Heat kernels on weighted manifolds and 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.

“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 space.” 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). Lapµ = ∂2/∂r2 + m(r) ∂/∂r + ψ-2 Lap_theta m(r) = (log S(r))’ = (n-1)(log g(r))’ + 2ψ’.

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

[2006] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009), 903–991; (2006).

[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.

[2007] Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Émery Ricci tensor, (2007). And see references therein.

[2009] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide. Academic Press, fourth edition, 2009.


[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):74–113, 2007.

[2008] Edward M. Fan, Topology of three-manifolds with positive P-scalar curvature. Proc. Amer. Math. Soc. 136 (2008), 3255–3261. [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, 71–74.

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

[2009] Y. Ollivier, Ricci Curvature of Markov Chains on Metric Spaces, J. Funct. Anal. 256 (2009), 810-864.

[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.

[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), 1203–1206.

[2010] Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran, Isoperimetric regions in the plane with density r^p, New York J. Math. 16 (2010), 31-51.

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

[2010] Pak Tung Ho,The structure of ϕ-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature. Math. Ann. 348 (2010), 319–332.

[2010] Emil Saucan, Curvature based triangulation of metric measure spaces.

[2010] Kevin Brighton, A Liouville-type theorem for smooth metric measure spaces.

[2011] Doan The Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys. 61 (2011), 1625-1629.

[2011] Ovidiu Munteanu and Jiaping Wang, Smooth metric measure spaces with non-negative curvature,, 2011. “As an application, we conclude steady Ricci solitons must be connected at infinity.”

[2011] Emil Saucan, A simple sampling method for metric measure spaces.  Cites Corwin et al. as well as my MwD.

[2011] Jürgen Jost and Shiping Liu, Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs.

[2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below,

[2011] Sean Howe, The Log-Convex Density Conjecture and vertical surface area in warped products, preprint (2011),

[2011] Tapio Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces.

[2011] David Bate and Gareth Speight, Differentiability, porosity and doubling in metric measure spaces,

[2011] Emanuel Milman, Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition,, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041–1078. [2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below,

[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.

[2011] Jasun Gong, Differentiability of Lipschitz functions on doubling metric measure spaces,

[2011] Siddhartha Gadgi and Manjunath Krishnapur, Lipschitz correspondence between metric measure spaces and random distance matrices,

[2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savaré, Density of Lipschitz functons and equivalence of weak gradients in metric measure spaces,

[2011] Ben Andrews and Lei Ni, Eignvalue comparison on Bakry-Emery manifolds,

[2011] Tapio Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditons of Sturm,

[2011] Wolfang Löhr, Equivalence of Gromov-Prohorov- and Gromov’s box-metric on the space of metric measure spaces,

[2011] Shouhei Honda, A weak second differentiable structure on rectifiable metric measure spaces,

[2012] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; (2010); see blog posts 1 and 2.

[2012] Yan-Hui Su and Hui-Chun Zhang. Rigidity of manifolds with Bakry-Emery Ricci curvature bounded below. Geometriae Dedicata 160 (2012), 1–11.

[2012] Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré, Heat flow and calculus on metric measure spaces with Ricci curvature bounded below – the compact case,

[2012] Nicola Gigli, On the differential structure of metric measure spaces and applications,

[2012] Luigi Ambrosio, Nicola Gigli, Andrea Mondino, Tapio Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with sigma-finite measure,

[2012] Karl-Theodor Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces,  “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.

[2012] David Bate [student of David Preiss at Warwick], Structure of measures in Lipschitz differentiability spaces.

[2012] Haizhong Li and Yong Wei, f-minimal surface and manifold with positive m-Bakry-Émery Ricci curvature,  Generalize to density Choi-Schoen theorem that minimal surfaces of fixed topological type in closed 3-manifold smoothly compact.

[2012] Christian Ketterer [student of Sturm], Ricci curvature bounds for warped products,

[2012] Zahra Sinaei,Harmonic maps on smooth metric measure spaces and their convergence,  [Good references.]

[2012] Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds,

[2012] Jose M. Espinar, Weighted stable CMC surfaces in manifolds with density,

[2012] Alessio Figalli, Emanuel Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones  Degenerate Wulff shape!

[2012] Xu Cheng, Tito Mejia, Detang Zhou, Stability and compactness for complete f-minimal surfaces, Cites Morgan [2005].

[2012] Alexandru Kristály and Shin-ichi Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications,

[2012] Ovidiu Munteanu and Jiaping Wang, Geometry of manifolds with densities,  Bochner, Laplacian, entropy; Euclidean vol growth, splitting.

[2012] Gang Liu, Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor, Comm. Anal. Geom., 2nd varn, splitting.

[2012] Jon Wolfson, Eigenvalue gap theorems for a class of non symmetric elliptic operators on convex domains, “The class of operators includes the Bakry-Emery laplacian.”

[2013] Matheus Vieira, Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature, Arch. Math. 101 (2013), 581–590.

[2013] Qin Huang, Qihua Ruan, Applications of Some Elliptic Equations in Riemannian Manifolds,  (Heintze-Karcher for m-Ricci, later by Batista-Cavalcante; thank and cite Ma.)

[2013] Fabio Cavalletti and Martin Huesmann, Existence and uniqueness of optimal transport maps,

[2013] Debora Impera and Michele Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces,

[2013] Mingliang Cai, On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature,

[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,  Good overview introduction.

[2013] Giuseppe Savaré, Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces,

[2013] Antonio Cañete, César Rosales, Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities, Good references.

[2013] Xavier Cabre, Xavier Ros-Oton, Joaquim Serra, Sharp isoperimetric inequalities via the ABP method,  Uses Figalli-Indrei degenerate Wulff shape to handle nonsmooth cones.

[2013] Marcio Batista, Heudson Mirandola [Brazil], A Sobolev-type inequality for submanifolds in weighted Riemannian Manifolds,

[2013] Jia-Yong Wu, L^p-Liouville theorems on complete smooth metric measure spaces,

[2013] Xu Cheng, Tito Mejia, Detang Zhou, Simons’ type equation for f-minimal hypersurfaces and applications,

[2013] Xu Cheng, Detang Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces,

[2013] Nicola Garofalo, Andrea Mondino, Li-Yau and Harnack type inequalities in RCD^*(K,N) metric measure spaces,

[2013] Haizhong Li, Yong Wei, Sharp diameter estimates for compact manifold with boundary,  Ruan generalized Cheng to m-Ricci; LW generalize to manifolds with boundary.

[2013] Ye-Lin Ou, On f-biharmonic maps and f-biharmonic submanifolds,, after Wei-Jun Lu, On f-Biharmonic maps between Riemannian manifolds , arXiv:1305.5478

[2013] Marcio Batista, Marcos P. Cavalcante, The Heintze-Karcher-Ros Inequality and Eigenvalue Estimates in Weighted Manifolds (Heintze-Karcher for m-Ricci, earlier by Huang Ruan.)

[2013] Aaron Naber, Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces,

[2013] Bingyu Song, Guofang Wei, Guoqiang Wu, Monotonicity Formulas for Bakry-Emery Ricci Curvature,

[2013] Xu Cheng, Detang Zhou, Stability properties and gap theorem for complete f-minimal hypersurfaces,

[2013] Matthew McGonagle, John Ross, The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space,

[2013] Bobo Hua, Martin Kell, Chao Xia, Harmonic functions on metric measure spaces,

[2013] Emanuel Milman and Liran Rotem, Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures, Also negative dimension.

[2013] Luigi Ambrosio, Andrea Mondino, Giuseppe Savaré, On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces,

[2013] Maree Jaramillo, Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below,

[2013] Shouhei Honda, Cheeger constant, p-Laplacian, and Gromov-Hausdorff convergence,

[2013] Bruno Colbois, Ahmad El Soufi, Alessandro Savo, Eigenvalues of the Laplacian on a compact manifold with density,

[2013] Alexander V. Kolesnikov, Emanuel Milman, Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary, Unified positive and negative dimension.

[2013] Fabio Cavalletti, Monge problem in metric measure spaces with Riemannian curvature-dimension condition,

[2013] Shin-ichi Ohta, (K,N)-convexity and the curvature-dimension condition for negative N,

[2013]  William Wylie, Sectional curvature for Riemannian manifolds with density,

[2013] Christian Ketterer, Cones over metric measure spaces and the maximal diameter theorem,

[2013] Katherine Castro and César Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results,

[2013] Gregory R. Chambers, Proof of the Log-Convex Density Conjecture,, J. Eur. Math. Soc. 21 (2019), 2301–2332.

[2013] M Rupert and E Woolgar, Bakry-Émery black holes, [2013] E Woolgar, Scalar-tensor gravitation and the Bakry-Emery-Ricci tensor, [2013] Nicola Gigli, Andrea Mondino, Giuseppe Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows,

[2013] Martin Kell, On Interpolation and Curvature via Wasserstein Geodesics, “The proof of the Borell–Brascamp–Lieb (BBL) inequality for Riemannian manifolds by Cordero-Erausquin-McCann-Schmuckenschläger [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.”

[2013] Katrin Fässler, Pekka Koskela, Enrico Le Donne. Nonexistence of quasiconformal maps between certain metric measure spaces,

[2013] Gregory J Galloway and Eric Woolgar, Cosmological singularities in Bakry-Émery spacetimes,

[2013] Guy C. David, Bi-Lipschitz Pieces between Manifolds,

[2013] Jeff Cheeger, Bruce Kleiner, Inverse limit spaces satisfying a Poincare inequality,

[2013] Jia-Yong Wu, Peng Wu, Heat Kernels on Smooth Metric Measure Spaces with Nonnegative Curvature,

[2014] Takashi Shioya, Metric measure limits of spheres and complex projective spaces, [2014]  Yashar Memarian, Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds, [apparently incorrect]

[2014] Christian Ketterer, Tapio Rajala, Failure of topological rigidity results for the measure contraction property,

[2014] César Rosales, Isoperimetric and stable sets for log-concave perturbations of Gaussian measures,

[2014] Ayato Mitsuishi, Current and measure homologies,

[2014] Karl-Theodor Sturm, Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions, “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.”

[2014] Yu Kitabeppu, Sajjad Lakzian, Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups,

[2014] Renjin Jiang, The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces,

[2014] Andrea Mondino, Aaron Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I,

[2014] Yi Li, Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature,

[2014] Jeffrey S. Case, Sun-Yung Alice Chang, On fractional GJMS operators,

[2014] Luigi Ambrosio, Andrea Pinamonti, Gareth Speight, Weighted Sobolev Spaces on Metric Measure Spaces,

[2014] Jia-Yong Wu, Peng Wu, Heat kernel on smooth metric measure spaces and applications,

[2014] Nicola Gigli, Bangxian Han, The continuity equation on metric measure spaces,

[2014] Nicola Gigli, Nonsmooth differential geometry – An approach tailored for spaces with Ricci curvature bounded from below,

[2014] Frédéric Bernicot, Thierry Coulhon, Dorothee FreyGradient estimates, Poincaré inequalities, De Giorgi property and their consequences,

[2014] Nicola Gigli, Bangxian Han, Independence on p of weak upper gradients on RCD spaces,

[2014] A. Barros, R. Batista, E. Ribeiro Jr., Bounds on volume growth of geodesic balls for Einstein warped products,

[2014] Erlend Grong, Anton Thalmaier, Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Parts I and II,,

[2014] David Bate, Sean Li, Characterizations of rectifiable metric spaces, [2014] Emanuel Milman, Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension,

[2014] Jeffrey S. Case, A notion of the weighted σ_k-curvature for manifolds with density,

[2014] Simone Di Marino, Sobolev and BV spaces on metric measure spaces via derivations and integration by parts,

[2014] Jing Mao, The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature,

[2014] Takashi Shioya, Metric measure geometry,

[2014] Lee Kennard, William Wylie, Positive weighted sectional curvature,

[2014] Antonio G. Ache, Micah W. Warren, Coarse Ricci curvature and the manifold learning problem,

[2014] Shouhei Honda, Elliptic PDEs on compact Ricci limit spaces and applications,

[2014] Karl-Theodor Sturm, Gradient Flows for Semiconvex Functions on Metric Measure Spaces – Existence, Uniqueness and Lipschitz Continuity,

[2014] Christian Ketterer, Obata’s rigidity theorem for metric measure spaces,

[2014] Jia-Yong Wu, Peng Wu, On L^p-Liouville property for smooth metric measure spaces,

[2014] Paul W.Y. Lee, On measure contraction property without Ricci curvature lower bound,

[2014] Dominique Bakry (IMT), François Bolley (LPMA), Ivan Gentil (ICJ), The Li-Yau inequality and applications under a curvature-dimension condition,

[2015] Fabio Cavalletti, Andrea Mondino, Measure rigidity of Ricci curvature lower bounds,

[2015] William Wylie, Some curvature pinching results for Riemannian manifolds with density,

[2015] Sérgio Mendonça, Splitting, parallel gradient and Bakry-Emery Ricci curvature,

[2015] Guy C. David, Lusin-type theorems for Cheeger derivatives on metric measure spaces,

[2015] Fabio Cavalletti, Andrea Mondino, Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds, Invent. Math. 208 (2017), no. 3, 803–849. Generalize Milman smooth to non-branching CD.

[2015] Yin Jiang, Hui-Chun Zhang, Sharp Spectral Gaps on Metric Measure Spaces,

[2015] Ezequiel Barbosa, Ben Sharp, Yong Wei, Smooth compactness of f-minimal hypersurfaces with bounded f-index,

[2015] M. Carmen Domingo-Juan, Vicente Miquel, Reilly’s type inequality for the Laplacian associated to a density related with shrinkers for MCF,

[2015] Vicente Miquel, Francisco Viñado-Lereu, The curve shortening problem associated to a density,

[2015] M. P. Cavalcante, J. Q. Oliveira, M. S. Santos, Compactness in weighted manifolds and applications, Results Math. (2015).

[2015]  Wyatt Boyer, Bryan Brown, Gregory R. Chambers, Alyssa Loving, Sarah Tammen, Isoperimetric Regions in ℝ^n with density r^p,, Anal. Geom. Metr. Spaces 4 (2016), 236–265.

[2015] Homare Tadano, Remark on a diameter bound for complete manifolds with positive Bakry-Émery Ricci curvature,

[2015] Fabio Cavalletti, Andrea Mondino, Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds,

[2015] Emanuel Milman, Harmonic Measures on the Sphere via Curvature-Dimension,

[2015] Christian Ketterer, On the geometry of metric measure spaces with variable curvature bounds,

[2015] William Wylie, A warped product version of the Cheeger-Gromoll splitting theorem,

[2015] Luigi Ambrosio, Matthias Erbar, Giuseppe Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces,

[2015] Emanuel Milman, Spectral Estimates, Contractions and Hypercontractivity,

[2015] Christian Ketterer, Evolution variational inequality and Wasserstein control in variable curvature context,

[2015] Luigi Ambrosio, Andrea Mondino, Giuseppe Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces,

[2015] Max Fathi, Yan Shu, Curvature and transport inequalities for Markov chains in discrete spaces,

[2015] Shouhei Honda, Spectral convergence under bounded Ricci curvature,

[2015] Bang-Xian Han, Conformal transformation on metric measure spaces,

[2015] Feng Du, Jing Mao, Qiaoling Wang, Chuanxi Wu, The Gagliardo-Nirenberg inequality on metric measure spaces,

[2015] Martin Kell, On Cheeger and Sobolev differentials in metric measure spaces,

[2015] Yuzhao Wang, Huaiqian Li, Lower Bound Estimates for The First Eigenvalue of The Weighted p-Laplacian on Smooth Metric Measure Space, [2015] Nicola Gigli, Guido de Philippis, From volume cone to metric cone in the non smooth setting,

[2015] Renjin Jiang, Huichun Zhang, Hamilton’s Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces,

[2016] Tomasz Adamowicz, Michał Gaczkowski, Przemysław Górka, Harmonic functions on metric measure spaces,

[2016] Shin-ichi Ohta, A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetric inequality [for Finsler manifolds with density],

[2016] Hui-Chun Zhang, Xi-Ping Zhu, Local Li-Yau’s estimates on metric measure spaces,

[2016] William Wylie, Dmytro Yeroshkin, On the geometry of Riemannian manifolds with density, dedicated to Frank Morgan,

[2016] Qintao Deng, Fernando Galaz-Garcia, Luis Guijarro, Michael Munn, Three-Dimensional Alexandrov spaces with positive or nonnegative Ricci curvature, “We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the ∗(K,N) sense.”

[2016] Erbar Matthias, Juillet Nicolas, Smoothing and non-smoothing via a flow tangent to the Ricci flow,

[2016] Karl-Theodor Sturm, Super-Ricci Flows for Metric Measure Spaces. I, [2016] Jeffrey S. Case, A weighted renormalized curvature for manifolds with density,

[2016] Yu Kitabeppu, A Bishop type inequality on metric measure spaces with Ricci curvature bounded below,

[2016] Xian-tao Huang, Noncompact RCD(0,N) spaces with linear volume growth,

[2016] Luigi Ambrosio, Federico Stra, Dario Trevisan, Weak and strong convergence of derivations and stability of flows with respect to MGH convergence,

[2016] Luigi Ambrosio, Nicola Gigli, Simone Di Marino, Perimeter as relaxed Minkowski content in metric measure spaces, [2016] Fabrice Baudoin, Daniel J. Kelleher, Poincaré Duality, Bakry-Émery Estimates and Isoperimetry on Fractals,

[2016] Ronen Eldan, James R. Lee, Joseph Lehec, Transport-entropy inequalities and curvature in discrete-space Markov chains,

[2016] Andrea Mondino, Guofang Wei, On the universal cover and the fundamental group of an RCD∗(K,N)-space,  Nice intro.

[2016] Luigi Ambrosio, Andrea Mondino, Gaussian-type Isoperimetric Inequalities in RCD(K,∞) probability spaces for positive K,  After Bakry-Émery.

[2016] Sajjad Lakzian, Zachary Mcguirk, Global Poincaré inequality on Graphs via Conical Curvature-Dimension Conditions,

[2016] Luigi Ambrosio, Shouhei Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below,

[2016] Max Grieshammer, Thomas Rippl, Partial orders on metric measure spaces,

[2016] Qi S Zhang, Meng Zhu, New Volume Comparison results and Applications to degeneration of Riemannian metrics,

[2016] A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, Some isoperimetric inequalities on ℝN with respect to weights |x|^α,

[2016] Lingzhong Zeng, The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces,

[2016] Martin Kell, Andrea Mondino, On the volume measure of non-smooth spaces with Ricci curvature bounded below,

[2016] Nicola Gigli, Enrico Pasqualetto, Behaviour of the reference measure on spaces under charts,

[2016] Sylvester Eriksson-Bique, Classifying Poincaré Inequalities and the local geometry of RNP-Differentiability Spaces, [2016] Vicente Miquel, Francisco Viñado-Lereu, Type I singularities in the curve shortening flow associated to a density,

[2016] Robert Haslhofer, Aaron Naber, Ricci Curvature and Bochner Formulas for Martingales,

[2016] Luis Guijarro, Jaime Santos-Rodríguez, On the isometry group of RCD∗(K,N)-spaces,

[2016] Shiping Liu, Florentin Münch, Norbert Peyerimhoff, Bakry-Emery curvature and diameter bounds on graphs,

[2016] Fabio Cavalletti, Andrea Mondino, Optimal maps in essentially non-branching spaces,

[2016] Gerardo Sosa, The isometry group of an RCD*-space is Lie,

[2016] Lukáš Malý, Nageswari Shanmugalingam, Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity,

[2016] Li Ma, Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers,

[2016] Shouhei Honda, Ricci curvature and Orientability,

[2016] Christian Ketterer, Andrea Mondino, Sectional and intermediate Ricci curvature lower bounds via Optimal Transport,

[2016] Nguyen Thac Dung, Nguyen Ngoc Khanh, Quoc Anh Ngô, Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces,

[2016] Nguyen Thac Dung, Kieu Thi Thuy Linh, Ninh Van Thu, Gradient estimates for some evolution equations on complete smooth metric measure spaces,

[2016] Jia-Yong Wu, Comparison geometry for integral Bakry-Émery Ricci tensor bounds,

[2016] Jia-Yong Wu, Liouville property for f-harmonic functions with polynomial growth,

[2016] Fabio Cavalletti, Andrea Mondino, Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds, For perimeter, not just Minkowski content.

[2016] Guy C. David, Bruce Kleiner, Rectifiability of planes and Alberti representations,

[2016] Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Balls Isoperimetric in ℝ^n with Volume and Perimeter Densities r^m and r^p,

[2016] Ushio Tanaka, Gromov’s Problem: Bound the Expansion Coefficient from below in terms of the Observable Diameter of a Metric Measure Space, and its Diameter Bounds,

[2016] Panu Lahti, Strong approximation of sets of finite perimeter in metric spaces, [2016] Nicola Gigli, Enrico Pasqualetto, Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces,

[2016] Panu Lahti, Strict and pointwise convergence of BV functions in metric spaces,

[2016] Fabio Cavalletti, Emanuel Milman, The Globalization Theorem for the Curvature Dimension Condition,

[2017] Hilário Alencar, Adina Rocha, The f-Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons,

[2017] Hui-Chun Zhang, Xi-Ping Zhu, Weyl’s law on RCD∗(K,N) metric measure spaces,

[2017] Jia-Cheng Huang, Hui-Chun Zhang, Sharp gradient estimate for heat kernels on RCD∗(K,N) metric measure spaces,

[2017] Luigi Ambrosio, Shouhei Honda, David Tewodrose, Short-time behavior of the heat kernel and Weyl’s law on RCD∗(K,N)-spaces,

[2017] Bang-Xian Han, Andrea Mondino, Angles between curves in metric measure spaces,

[2017] Davide Barilari, Luca Rizzi, Sharp measure contraction property for generalized H-type Carnot groups,

[2017] Chang-Yu Guo, Harmonic mappings between singular metric spaces,

[2017] Ángel Arroyo, José G. Llorente, A priori Hölder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces,

[2017] Luigi Ambrosio, Shouhei Honda, Local spectral convergence in RCD∗(K,N) spaces,

[2017] Panu Lahti, Quasiopen sets, bounded variation and lower semicontinuity in metric spaces,

[2017] Kohei Suzuki, Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions,

[2017] Manuel Ritoré, Jesús Yepes Nicolás, Brunn-Minkowski inequalities in product metric measure spaces,

[2017] Nicola Gigli, Chiara Rigoni, Recognizing the flat torus among RCD∗(0,N) spaces via the study of the first cohomology group,

[2017] Hiroki Nakajima, The maximum of the 1-measurement of a metric measure space,

[2017] Nicola Gigli, Chiara Rigoni, A note about the strong maximum principle on RCD space,

[2017] Luigi Ambrosio, Shouhei Honda, Jacobus W. Portegies, Continuity of nonlinear eigenvalues in CD(K,∞) spaces with respect to measured Gromov-Hausdorff convergence,

[2017] Jia-Yong Wu, Myers’ type theorem with the Bakry-Émery Ricci tensor,

[2017] Lee Kennard, William Wylie, Dmytro Yeroshkin, The weighted connection and sectional curvature for manifolds with density,

[2017] Eric Woolgar, William Wylie, Curvature-dimension bounds for Lorentzian splitting theorems,

[2017] Guido De Philippis, Nicola Gigli, Non-collapsed spaces with Ricci curvature bounded from below,

[2017] Yu Kitabeppu, A sufficient condition to a regular set of positive measure on RCD spaces,

[2017] Nicola Gigli, Christian Ketterer, Kazumasa Kuwada, Shin-ichi Ohta, Rigidity for the spectral gap on RCD(K,∞)-spaces,

[2017] Kohei Suzuki, Convergence of Non-Symmetric Diffusion Processes on RCD spaces,

[2017] Bang-Xian Han, Ricci tensor on smooth metric measure space with boundary,

[2017] Xian-Tao Huang, An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth,

[2017] Bang-Xian Han, Characterizations of monotonicity of vector fields on metric measure space,

[2017] Helmer Hoppe, Jun Masamune, Stefan Neukamm, H-compactness of elliptic operators on weighted Riemannian Manifolds,

[2017] Luca Rizzi, A counterexample to gluing theorems for MCP metric measure spaces,

[2017] Willian Isao Tokura, Levi Adriano, Changyu Xia, The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces,

[2017] Vitali Kapovitch, Christian Ketterer, RCD meets CAT,

[2017] Timo Schultz, Existence of optimal transport maps in very strict CD(K,∞) -spaces,

[2017] Luigi Ambrosio, Elia Bruè, Dario Trevisan, Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and RCD(K,∞) spaces,

[2017] Cong Hung Mai, Rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds,

[2017] Raphael Bouyrie, Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry–Ledoux isoperimetric comparison Theorem,

[2017] Matthias Erbar, Karl-Theodor Sturm, Rigidity of cones with bounded Ricci curvature,

[2018] Hiroki Nakajima, Takashi Shioya, Isoperimetric rigidity and distributions of 1-Lipschitz functions,

[2018] Antonio Bueno, Translating solitons of the mean curvature flow in the space ℍ2×ℝ,

[2018] Elia Bruè, Daniele Semola, Regularity of Lagrangian flows over RCD∗(K,N) spaces,

[2018] Sebastiano Don, Davide Vittone, A compactness result for BV functions in metric spaces,

[2018] Debora Impera, Michele Rimoldi, Alessandro Savo, Index and first Betti number of f-minimal hypersurfaces and self-shrinkers,

[2018] Daisuke Kazukawa, A new condition for convergence of energy functionals and stability of lower Ricci curvature bound,

[2018] Elia Bruè, Daniele Semola, Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows,

[2018] Shouhei Honda, Bakry-Émery conditions on almost smooth metric measure spaces,

[2018] Ilaria Mondello (LAMA), J. Bertrand (IMT), C Ketterer, T. Richard (LAMA), Stratified spaces and synthetic Ricci curvature bounds,

[2018] Antoni Kijowski, Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure, “We conclude with a remarkable observation that strongly harmonic functions in R^n possess the mean value property with respect to infinitely many weight functions obtained from a given weight.”

[2018] Angelo Alvino, Friedemann Brock, Francesco Chiacchio, Anna Mercaldo, Maria Rosaria Posteraro, The isoperimetric problem for a class of non-radial weights and applications,

[2018] Jhovanny Muñoz Posso, A generalization of Sobolev trace inequality and Escobar-Riemann mapping type problem on smooth metric measure spaces, [part of PhD thesis under Fernando Codá´Marques]

[2018] Jeff Cheeger, Wenshuai Jiang, Aaron Naber, Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below,

[2018] Ana Hurtado, Vicente Palmer, César Rosales, Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights, [2018] Katrin Fässler, Tuomas Orponen, Metric currents and the Poincaré inequality,

[2018] Danka Lučić, Enrico Pasqualetto, Infinitesimal Hilbertianity of weighted Riemannian manifolds,

[2018] Chris Connell, Xianzhe Dai, Jesús Núñez-Zimbrón, Raquel Perales, Pablo Suárez-Serrato, Guofang Wei, Maximal volume entropy rigidity for ∗(−(N−1),N) spaces,

[2018] Jaime Santos-Rodríguez, Invariant measures and lower Ricci curvature bounds,

[2018] Fabio Cavalletti, Flavia Santarcangelo, Isoperimetric inequality under Measure-Contraction property, Generalize Levy-Gromov-Milman-CavallettiMondino to MCP spaces.

[2018] L. M. Chasman, Jeffrey J Langford, On Clamped Plates with Log-Convex Density,

[2018] Panu Lahti, A sharp Leibniz rule for BV functions in metric spaces,

[2018] Kazumasa Kuwada, Xiang-Dong Li, Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces,

[2018] Simone Di Marino, Nicola Gigli, Enrico Pasqualetto, Elefterios Soultanis, Infinitesimal Hilbertianity of locally CAT(κ)-spaces,


[2019] Vitali Kapovitch, Christian Ketterer, Weakly noncollapsed RCD spaces with upper curvature bounds,

[2019] Bang-XIan Han, Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds,

[2019] Nicola Gigli, Riemann curvature tensor on spaces and possible applications,

[2019] Friedemann Brock, Francesco Chiacchio, Some weighted isoperimetric problems on ℝ^N+ with stable half balls have no solutions,

[2019] Adriano Cavalcante Bezerra, Changyu Xia, Sharp Estimates for the First Eigenvalues of the Bi-drifting [squared] Laplacian,

[2019] Giulia Luise, Giuseppe Savaré, Contraction and regularizing properties of heat flows in metric measure spaces,

[2019] Shin-ichi Ohta, Asuka Takatsu, Equality in the logarithmic Sobolev inequality,

[2019] Shouhei Honda, New differential operator and non-collapsed RCD spaces,

[2019] Timo Schultz, Equivalent definitions of very strict CD(K,N) -spaces,

[2019] Davide Barilari, Luca Rizzi, Bakry-Émery curvature and model spaces in sub-Riemannian geometry,

[2019] Mathias Braun, Karen Habermann, Karl-Theodor Sturm, Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds,

[2019] Gioacchino Antonelli, Elia Bruè, Daniele Semola, Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces,

[2019] Vitali Kapovitch, Andrea Mondino, On the topology and the boundary of N-dimensional RCD(K,N) spaces,

[2019] Shouhei Honda, Ilaria Mondello, Sphere theorems for RCD spaces,

[2019] Angelo Alvino, Friedemann Brock, Francesco Chiacchio, Anna Mercaldo, Maria Rosaria Posteraro, Some isoperimetric inequalities with respect to monomial weights,

[2019] Bang-Xian Han, Karl-Theodor Sturm, Curvature-dimension conditions for diffusions under time change,

[2019] Ana Hurtado, Vicente Palmer, César Rosales, Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds,

[2019] Batu Güneysu, Max von Renesse, Molecules as metric measure spaces with Kato-bounded Ricci curvature,

[2019] Christian Ketterer, The Heintze-Karcher inequality for metric measure spaces,

[2019] Vitali Kapovitch, Martin Kell, Christian Ketterer, On the structure of RCD spaces with upper curvature bounds,

[2019] Abdolhakim Shouman, Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi‑drifting Laplacian, Annals of Global Analysis and Geometry (2019) 55:805–817 1 3

[2020] Adson Meira · Rosivaldo Antonio Gonçalves, On the space of f-minimal surfaces with bounded f-index in weighted smooth metric spaces, manuscripta math. 162, 559–563 (2020)

[2020] Kazuhiro Kuwae, Yohei Sakurai, Comparison geometry of manifolds with boundary under lower N-weighted Ricci curvature bounds with ε-range,

[2021] Florian Johne, Sobolev inequalities on manifolds with nonnegative Bakry-Émery Ricci curvature,

[2021] Homare Tadanoa, Some compactness theorems via m-Bakry–Émery and m-modified Ricci curvatures with negative m, Differential Geometry and its Applications 75 (2021), 101-720

[2021] Allan G. Freitasa, Henrique F. de Lima, Eraldo A. Lima Jr., Márcio S. Santosa, Submanifolds immersed in Riemannian spaces endowed with a Killing vector field: Nonexistence and rigidity, Differential Geometry and its Applications 75 (2021) 101-714

[2021] Fabio Cavalletti, Davide Manini, Isoperimetric inequality in noncompact MCP spaces,


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

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

[B5] F. Barthe and B. Maurey, Some remarks on isoperimetry of gaussian type, Ann. Inst. H. Poincaré Prob. Stat. 36, (2000), 419-434.

[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

[B8] Sergei Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Prob. 24, 1996, 35-48.

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

[B10] Sergei Bobkov and Christian Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129, 1997.

[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.

[D] E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. d’Analyse Math. 58 (1992) 99-119.

[E] A. Ehrhard, Inégalités isopérimetriques et intégrales de Dirichlet gaussienes, Ann. Sci. Ecole Norm. Sup. 17 (1984), 317-332.

[FOT] M. Fukushima, Y. Oshima, M. Takeda, “Dirichlet forms and symmetric Markov processes”, Studies in Mathematics 19, De Gruyter, 1994.

[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.”

[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.

[L0] Michel Ledoux, Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes, C. F. Acad. Sci. Paris 206 (1988), 79-82. Houdré tells me this is first observation that some log-Sobolev follows from isoperimetry, though not Gross version.

[L1] Michel Ledoux, The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, 89, Amer. Math. Soc., 2001.

[L2] M. Ledoux. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6), 9(2):305–366, 2000.

[L3] M. Ledoux. A simple analytic proof of an inequality by P. Buser. Proc. Amer. Math. 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]).

[Lo] J. Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comm. Math. Helv. 78, 2003, 865-883.

[McK] H. P. McKean, Geometry of differential space, Ann. Prob. 1 (1973), 197-206.

[N] E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973), 211-227.

[O] Shin-ichi Ohta,Optimal transport and Ricci curvature in Finsler geometry, Advanced Studies in Pure Mathematics  (2010), 1–20.

[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-Weitzenböck 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]).

[Q2] Zhongmin Qian, Estimates for weighted volumes and applications, Quart. J. Math. 48 (1997), 235-242.

[Stam] Stam, A. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2 (1959), 101–112. 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].

[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.

[RBCM] César Rosales, Vincent Bayle, Antonio Cañete, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Varn. PDE (2007).

[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

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