#### Volume 21, issue 1 (2017)

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Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

### Fabio Cavalletti and Andrea Mondino

Geometry & Topology 21 (2017) 603–645
##### Bibliography
 1 L Ambrosio, N Gigli, A Mondino, T Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ–finite measure, Trans. Amer. Math. Soc. 367 (2015) 4661 MR3335397 2 L Ambrosio, N Gigli, G Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014) 289 MR3152751 3 L Ambrosio, N Gigli, G Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014) 1405 MR3205729 4 L Ambrosio, N Gigli, G Savaré, Bakry–Émery curvature–dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015) 339 MR3298475 5 L Ambrosio, A Mondino, G Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, preprint (2015) arXiv:1509.07273 6 L Ambrosio, A Mondino, G Savaré, On the Bakry–Émery condition, the gradient estimates and the local-to-global property of RCD∗(K,N) metric measure spaces, J. Geom. Anal. 26 (2016) 24 MR3441502 7 K Bacher, K T Sturm, Localization and tensorization properties of the curvature–dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28 MR2610378 8 D Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, from: "Lectures on probability theory" (editor P Bernard), Lecture Notes in Math. 1581, Springer (1994) 1 MR1307413 9 D Bakry, Z Qian, Some new results on eigenvectors via dimension, diameter, and Ricci curvature, Adv. Math. 155 (2000) 98 MR1789850 10 S G Bobkov, C Houdré, Isoperimetric constants for product probability measures, Ann. Probab. 25 (1997) 184 MR1428505 11 C Borell, Convex set functions in d–space, Period. Math. Hungar. 6 (1975) 111 MR0404559 12 H J Brascamp, E H Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976) 366 MR0450480 13 F Cavalletti, Decomposition of geodesics in the Wasserstein space and the globalization problem, Geom. Funct. Anal. 24 (2014) 493 MR3192034 14 F Cavalletti, A Mondino, Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds, Invent. Math. (2016) 15 F Cavalletti, A Mondino, Optimal maps in essentially non-branching spaces, Commun. Contemp. Math. (2017) 16 F Cavalletti, K T Sturm, Local curvature–dimension condition implies measure-contraction property, J. Funct. Anal. 262 (2012) 5110 MR2916062 17 I Chavel, Eigenvalues in Riemannian geometry, 115, Academic Press (1984) MR768584 18 I Chavel, Isoperimetric inequalities: differential geometric and analytic perspectives, 145, Cambridge University Press (2001) MR1849187 19 J Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, from: "Problems in analysis", Princeton Univ. Press (1970) 195 MR0402831 20 J Cheeger, T H Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189 MR1405949 21 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 MR1484888 22 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000) 13 MR1815410 23 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000) 37 MR1815411 24 T H Colding, A Naber, Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. 176 (2012) 1173 MR2950772 25 D Cordero-Erausquin, R J McCann, M Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001) 219 MR1865396 26 M Erbar, K Kuwada, K T Sturm, On the equivalence of the entropic curvature–dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015) 993 MR3385639 27 H Federer, W H Fleming, Normal and integral currents, Ann. of Math. 72 (1960) 458 MR0123260 28 N Gigli, M Ledoux, From log Sobolev to Talagrand : a quick proof, Discrete Contin. Dyn. Syst. 33 (2013) 1927 MR3002735 29 N Gigli, A Mondino, G Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. 111 (2015) 1071 MR3477230 30 M Gromov, V D Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987) 263 MR901393 31 E Hebey, Sobolev spaces on Riemannian manifolds, 1635, Springer (1996) MR1481970 32 S Honda, Cheeger constant, p–Laplacian, and Gromov–Hausdorff convergence, preprint (2014) arXiv:1310.0304v3 33 Y Jiang, H C Zhang, Sharp spectral gaps on metric measure spaces, Calc. Var. Partial Differential Equations 55 (2016) MR3449924 34 R Kannan, L Lovász, M Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995) 541 MR1318794 35 C Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. 103 (2015) 1228 MR3333056 36 C Ketterer, Obata’s rigidity theorem for metric measure spaces, Anal. Geom. Metr. Spaces 3 (2015) 278 MR3403434 37 B Klartag, Needle decomposition in Riemannian geometry, preprint (2014) arXiv:1408.6322 38 M Ledoux, The geometry of Markov diffusion generators : probability theory, Ann. Fac. Sci. Toulouse Math. 9 (2000) 305 MR1813804 39 M Ledoux, Spectral gap, logarithmic Sobolev constant, and geometric bounds, from: "Surveys in differential geometry, IX : Eigenvalues of Laplacians and other geometric operators" (editors A Grigor’yan, S T Yau), Surv. Differ. Geom. 9, International Press (2004) 219 MR2184990 40 P Li, S T Yau, Estimates of eigenvalues of a compact Riemannian manifold, from: "Geometry of the Laplace operator" (editors R Osserman, A Weinstein), Proc. Sympos. Pure Math. XXXVI, Amer. Math. Soc. (1980) 205 MR573435 41 A Lichnerowicz, Géométrie des groupes de transformations, III, Dunod (1958) MR0124009 42 J Lott, C Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (2007) 311 MR2311627 43 J Lott, C Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009) 903 MR2480619 44 L Lovász, M Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms 4 (1993) 359 MR1238906 45 A M Matei, First eigenvalue for the p–Laplace operator, Nonlinear Anal. 39 (2000) 1051 MR1735181 46 V G Maz’ja, Sobolev spaces, Springer (1985) MR817985 47 E Milman, On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (2009) 1 MR2507637 48 E Milman, Sharp isoperimetric inequalities and model spaces for the curvature–dimension–diameter condition, J. Eur. Math. Soc. 17 (2015) 1041 MR3346688 49 E Milman, L Rotem, Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures, Adv. Math. 262 (2014) 867 MR3228444 50 C E Mueller, F B Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n–sphere, J. Funct. Anal. 48 (1982) 252 MR674060 51 A Naber, D Valtorta, Sharp estimates on the first eigenvalue of the p–Laplacian with negative Ricci lower bound, Math. Z. 277 (2014) 867 MR3229969 52 S i Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009) 211 MR2546027 53 F Otto, C Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000) 361 MR1760620 54 L E Payne, H F Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960) 286 MR0117419 55 A Profeta, The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds, Potential Anal. 43 (2015) 513 MR3430465 56 T Rajala, Failure of the local-to-global property for CD(K,N) spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 15 (2016) 45 MR3495420 57 T Rajala, K T Sturm, Non-branching geodesics and optimal maps in strong CD(K,∞)–spaces, Calc. Var. Partial Differential Equations 50 (2014) 831 MR3216835 58 R Schoen, S T Yau, Lectures on differential geometry, I, International Press (1994) MR1333601 59 K T Sturm, On the geometry of metric measure spaces, I, Acta Math. 196 (2006) 65 MR2237206 60 K T Sturm, On the geometry of metric measure spaces, II, Acta Math. 196 (2006) 133 MR2237207 61 D Valtorta, Sharp estimate on the first eigenvalue of the p–Laplacian, Nonlinear Anal. 75 (2012) 4974 MR2927560 62 C Villani, Optimal transport: old and new, 338, Springer (2009) MR2459454 63 Y Z Wang, H Q Li, Lower bound estimates for the first eigenvalue of the weighted p–Laplacian on smooth metric measure spaces, Differential Geom. Appl. 45 (2016) 23 MR3457386 64 J Y Wu, E M Wang, Y Zheng, First eigenvalue of the p–Laplace operator along the Ricci flow, Ann. Global Anal. Geom. 38 (2010) 27 MR2657841 65 J Q Zhong, H C Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984) 1265 MR794292