Stability estimates for the sharp spectral gap bound under a curvature-dimension condition
[Estimations du trou spectral optimal sous une condition de courbure-dimension]
Fathi, Max1 ; Gentil, Ivan2 ; Serres, Jordan3
1 Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques Louis Lions & Laboratoire de Probabilités Statistique et Modélisation, 8 place Aurélie Nemours, F-75013 Paris (France)
2 Institut Camille Jordan, Umr Cnrs 52065, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex (France)
3 CREST, ENSAE Paris, Institut Polytechnique de Paris 5 avenue Le Chatelier, 91120 Palaiseau (France)
Annales de l'Institut Fourier, Online first, 35 p.

Nous étudions la stabilité du trou spectral pour des espaces métriques mesurés vérifiant un critère de courbure dimension. Notre résultat principal, nouveau même dans le cadre d’un espace lisse, est une estimation quantitative montrant que si le trou spectral d’un espace RCD(N-1,N) est presque minimal alors l’image par la fonction propre associée au trou spectral de la mesure de référence est proche d’une loi Beta. La preuve combine des estimations de la fonction propre par le biais d’une nouvelle inégalité fonctionnelle de type L 1 dans un espace RCD et la méthode de Stein pour comparer des lois. Nous obtenons aussi des bornes presque optimales pour des espaces ayant une dimension intrinsèque infinie ou bien négative.

We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an RCD(N-1,N) space is almost minimal, then the pushforward of the measure by an eigenfunction associated with the spectral gap is close to a Beta distribution. The proof combines estimates on the eigenfunction obtained via a new L 1 -functional inequality for RCD spaces with Stein’s method for distribution approximation. We also derive analogous, almost sharp, estimates for infinite and negative values of the dimension parameter.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3608
Classification : 60B05, 46-XX
Keywords: Spectral gap, Poincaré inequalities, RCD spaces, Curvature-dimension condition, Stein method.
Mot clés : Trou spectral, inégalité de Poincaré, espaces RCD, condition de courbure-dimension, méthode de Stein.
Fathi, Max 1 ; Gentil, Ivan 2 ; Serres, Jordan 3

1 Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques Louis Lions & Laboratoire de Probabilités Statistique et Modélisation, 8 place Aurélie Nemours, F-75013 Paris (France)
2 Institut Camille Jordan, Umr Cnrs 52065, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex (France)
3 CREST, ENSAE Paris, Institut Polytechnique de Paris 5 avenue Le Chatelier, 91120 Palaiseau (France) 
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Fathi, Max; Gentil, Ivan; Serres, Jordan. Stability estimates for the sharp spectral gap bound under a curvature-dimension condition. Annales de l'Institut Fourier, Online first, 35 p.

[1] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., Volume 195 (2014) no. 2, pp. 289-391 | DOI | MR | Zbl

[2] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., Volume 163 (2014) no. 7, pp. 1405-1490 | DOI | MR | Zbl

[3] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab., Volume 43 (2015) no. 1, pp. 339-404 | DOI | MR | Zbl

[4] Anderson, Michael T. Metrics of positive Ricci curvature with large diameter, Manuscr. Math., Volume 68 (1990) no. 4, pp. 405-415 | DOI | MR | Zbl

[5] Aubry, Erwann Pincement sur le spectre et le volume en courbure de Ricci positive, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005) no. 3, pp. 387-405 | DOI | Numdam | MR | Zbl

[6] Bakry, Dominique; Émery, Michel Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Math. Acad. Sci. Paris, Volume 301 (1985) no. 8, pp. 411-413 | MR | Zbl

[7] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014, xx+552 pages | DOI | MR | Zbl

[8] Bakry, Dominique; Gentil, Ivan; Scheffer, Grégory Sharp Beckner-type inequalities for Cauchy and spherical distributions, Stud. Math., Volume 251 (2020) no. 3, pp. 219-245 | DOI | MR | Zbl

[9] Bertrand, Jérôme Pincement spectral en courbure de Ricci positive, Comment. Math. Helv., Volume 82 (2007) no. 2, pp. 323-352 | DOI | MR | Zbl

[10] Bertrand, Jérôme; Fathi, Max Stability of eigenvalues and observable diameter in RCD (1,∞) spaces, J. Geom. Anal., Volume 32 (2022) no. 11, 270, 38 pages | DOI | MR | Zbl

[11] Bertrand, Jérôme; Ketterer, Christian; Mondello, Ilaria; Richard, Thomas Stratified spaces and synthetic Ricci curvature bounds, Ann. Inst. Fourier, Volume 71 (2021) no. 1, pp. 123-173 | DOI | Numdam | MR | Zbl

[12] Cattiaux, Patrick; Guillin, Arnaud; Roberto, Cyril Poincaré inequality and the L p convergence of semi-groups, Electron. Commun. Probab., Volume 15 (2010), pp. 270-280 | DOI | MR | Zbl

[13] Cavalletti, Fabio; Mondino, Andrea; Semola, Daniele Quantitative Obata’s Theorem (2022) (https://arxiv.org/abs/1910.06637)

[14] Chatterjee, Sourav A short survey of Stein’s method, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. IV, Kyung Moon Sa, 2014, pp. 1-24 | MR | Zbl

[15] Cheng, Shiu Yuen Eigenvalue comparison theorems and its geometric applications, Math. Z., Volume 143 (1975) no. 3, pp. 289-297 | DOI | MR | Zbl

[16] Croke, Christopher B. An eigenvalue pinching theorem, Invent. Math., Volume 68 (1982) no. 2, pp. 253-256 | DOI | MR | Zbl

[17] Döbler, Christian Stein’s method of exchangeable pairs for the beta distribution and generalizations, Electron. J. Probab., Volume 20 (2015), 109, 34 pages | DOI | MR | Zbl

[18] Erbar, Matthias; Kuwada, Kazumasa; Sturm, Karl-Theodor On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math., Volume 201 (2015) no. 3, pp. 993-1071 | DOI | MR | Zbl

[19] Erbar, Matthias; Sturm, Karl-Theodor Rigidity of cones with bounded Ricci curvature, J. Eur. Math. Soc., Volume 23 (2021) no. 1, pp. 219-235 | DOI | MR | Zbl

[20] Eskenazis, Alexandros; Ivanisvili, Paata Sharp growth of the Ornstein–Uhlenbeck operator on Gaussian tail spaces, Isr. J. Math., Volume 253 (2023) no. 1, pp. 469-485 | DOI | MR | Zbl

[21] Gentil, Ivan; Zugmeyer, Simon A family of Beckner inequalities under various curvature-dimension conditions, Bernoulli, Volume 27 (2021) no. 2, pp. 751-771 | DOI | MR | Zbl

[22] Gigli, Nicola; Mondino, Andrea; Savaré, Giuseppe Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc., Volume 111 (2015) no. 5, pp. 1071-1129 | DOI | MR | Zbl

[23] Goldstein, Larry; Reinert, Gesine Stein’s method for the beta distribution and the Pólya–Eggenberger urn, J. Appl. Probab., Volume 50 (2013) no. 4, pp. 1187-1205 | DOI | MR | Zbl

[24] Honda, Shouhei Bakry–Émery conditions on almost smooth metric measure spaces, Anal. Geom. Metr. Spaces, Volume 6 (2018) no. 1, pp. 129-145 | DOI | MR | Zbl

[25] Honda, Shouhei; Mondello, Ilaria Sphere theorems for RCD and stratified spaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 22 (2021) no. 2, pp. 903-923 | DOI | MR | Zbl

[26] Ilias, Saïd Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier, Volume 33 (1983) no. 2, pp. 151-165 | DOI | Numdam | MR | Zbl

[27] Jiang, Yin; Zhang, Hui-Chun Sharp spectral gaps on metric measure spaces, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 1, 14, 14 pages | DOI | MR | Zbl

[28] Ketterer, Christian Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl., Volume 103 (2015) no. 5, pp. 1228-1275 | DOI | MR | Zbl

[29] Ketterer, Christian Obata’s rigidity theorem for metric measure spaces, Anal. Geom. Metr. Spaces, Volume 3 (2015) no. 1, pp. 278-295 | DOI | MR | Zbl

[30] Klartag, Bo’az Needle decompositions in Riemannian geometry, Memoirs of the American Mathematical Society, 249, American Mathematical Society, 2017 no. 1180, v+77 pages | DOI | MR | Zbl

[31] Kolesnikov, Alexander V.; Milman, Emanuel Brascamp–Lieb-type inequalities on weighted Riemannian manifolds with boundary, J. Geom. Anal., Volume 27 (2017) no. 2, pp. 1680-1702 | DOI | MR | Zbl

[32] Kolesnikov, Alexander V.; Milman, Emanuel Poincaré and Brunn–Minkowski inequalities on the boundary of weighted Riemannian manifolds, Am. J. Math., Volume 140 (2018) no. 5, pp. 1147-1185 | DOI | MR | Zbl

[33] Lott, John; Villani, Cédric Ricci curvature for metric-measure spaces via optimal transport, Ann. Math., Volume 169 (2009) no. 3, pp. 903-991 | DOI | MR | Zbl

[34] Mai, Cong Hung On Riemannian manifolds with positive weighted Ricci curvature of negative effective dimension, Kyushu J. Math., Volume 73 (2019) no. 1, pp. 205-218 | DOI | MR | Zbl

[35] Meckes, Elizabeth On the approximate normality of eigenfunctions of the Laplacian, Trans. Am. Math. Soc., Volume 361 (2009) no. 10, pp. 5377-5399 | DOI | MR | Zbl

[36] Mendel, Manor; Naor, Assaf Nonlinear spectral calculus and super-expanders, Publ. Math., Inst. Hautes Étud. Sci., Volume 119 (2014), pp. 1-95 | DOI | Numdam | MR | Zbl

[37] Meyer, Paul-André Transformations de Riesz pour les lois gaussiennes, Seminar on probability, XVIII (Lecture Notes in Mathematics), Volume 1059, Springer, 1984, pp. 179-193 | DOI | Numdam | MR | Zbl

[38] Milman, Emanuel Beyond traditional curvature-dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension, Trans. Am. Math. Soc., Volume 369 (2017) no. 5, pp. 3605-3637 | DOI | MR | Zbl

[39] Naor, Assaf; Schechtman, Gideon Remarks on non linear type and Pisier’s inequality, J. Reine Angew. Math., Volume 552 (2002), pp. 213-236 | DOI | MR | Zbl

[40] Obata, Morio Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, Volume 14 (1962), pp. 333-340 | DOI | MR | Zbl

[41] Ohta, Shin-ichi (K,N)-convexity and the curvature-dimension condition for negative N, J. Geom. Anal., Volume 26 (2016) no. 3, pp. 2067-2096 | DOI | MR | Zbl

[42] Petersen, Peter On eigenvalue pinching in positive Ricci curvature, Invent. Math., Volume 138 (1999) no. 1, pp. 1-21 | DOI | MR | Zbl

[43] Profeta, Angelo The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds, Potential Anal., Volume 43 (2015) no. 3, pp. 513-529 | DOI | MR | Zbl

[44] Ross, Nathan Fundamentals of Stein’s method, Probab. Surv., Volume 8 (2011), pp. 210-293 | DOI | MR | Zbl

[45] Savaré, Giuseppe Self-improvement of the Bakry–Émery condition and Wasserstein contraction of the heat flow in RCD(K,∞) metric measure spaces, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 4, pp. 1641-1661 | DOI | MR | Zbl

[46] Scheffer, Grégory Local Poincaré inequalities in non-negative curvature and finite dimension, J. Funct. Anal., Volume 198 (2003) no. 1, pp. 197-228 | DOI | MR | Zbl

[47] Stein, Charles A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, University of California Press (1972), pp. 583-602 | MR | Zbl

[48] Stein, Charles Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7, Institute of Mathematical Statistics, 1986, iv+164 pages | DOI | MR | Zbl

[49] Sturm, Karl-Theodor On the geometry of metric measure spaces. I, Acta Math., Volume 196 (2006) no. 1, pp. 65-131 | DOI | MR | Zbl

[50] Sturm, Karl-Theodor On the geometry of metric measure spaces. II, Acta Math., Volume 196 (2006) no. 1, pp. 133-177 | DOI | MR | Zbl

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