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Tytuł artykułu

Obata’s Rigidity Theorem for Metric Measure Spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian of a sufficiently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher order eigenvalues.
Słowa kluczowe
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-03-01
zaakceptowano
2015-08-25
online
2015-10-01
Twórcy
  • University of Freiburg Freiburg, Germany, Germany
Bibliografia
  • [1] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Bakry-Emery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015), no. 1, 339–404
  • [2] , Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014), no. 2, 289–391.
  • [3] , Metric measure spaces with Riemannian Ricci curvature bounded from below, DukeMath. J. 163 (2014), no. 7, 1405– 1490.
  • [4] Anders Björn and Jana Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich, 2011.
  • [5] Dominique Bakry and Zhongmin Qian, Some new results on eigenvectors via dimension, diameter, and Ricci curvature, Adv. Math. 155 (2000), no. 1, 98–153.
  • [6] Kathrin Bacher and Karl-Theodor Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010), no. 1, 28–56.
  • [7] Fabio Cavalletti and Andrea Mondino, Sharp geometric and functional inequalities in metric measure spaces with lower ricci curvature bounds, http://arxiv.org/abs/1505.02061.
  • [8] Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm, On the equivalence of the Entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015), no. 3, 993–1071
  • [9] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, extended ed., de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011.
  • [10] Nicola Gigli, On the differential structure of metric measure spaces and applications, Providence, Rhode Island: American Mathematical Society, 2015.
  • [11] Nicola Gigli and Andrea Mondino, A PDE approach to nonlinear potential theory in metric measure spaces, J. Math. Pures Appl. (9) 100 (2013), no. 4, 505–534.
  • [12] Alexander Grigor’yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI, 2009.
  • [13] Nicola Gigli, Tapio Rajala, and Karl-Theodor Sturm, Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal., to appear.
  • [14] Yin Jiang and Hui-Chun Zhang, Sharp spectral gaps on metric measure spaces, http://arxiv.org/abs/1503.00203.
  • [15] Christian Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J.Math. Pures Appl. (9) 103 (2015), no. 5, 1228–1275.
  • [16] , Ricci curvature bounds for warped products, J. Funct. Anal. 265 (2013), no. 2, 266–299.
  • [17] Pawel Kröger, On the spectral gap for compact manifolds, J. Differential Geom. 36 (1992), no. 2, 315–330.
  • [18] John Lott and Cédric Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (2007), no. 1, 311–333.
  • [19] M. Obata, Certain conditions for a riemannian manifold to be isometric with a sphere, J. Math. Soc. Jpn. 14 (1962), no. 14, 333–340. [Crossref]
  • [20] Shin-ichi Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), no. 4, 805–828.
  • [21] , Products, cones, and suspensions of spaces with themeasure contraction property, J. Lond.Math. Soc. (2) 76 (2007), no. 1, 225–236.
  • [22] A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), no. 1, 123–148.
  • [23] Zhongmin Qian, Hui-Chun Zhang, and Xi-Ping Zhu, Sharp spectral gap and Li-Yau’s estimate on Alexandrov spaces, Math. Z. 273 (2013), no. 3-4, 1175–1195.
  • [24] Giuseppe Savaré, Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,1) metric measure spaces, Discrete Contin. Dyn. Syst. 34 (2014), no. 4, 1641–1661.
  • [25] Karl-Theodor Sturm, Gradient flows for semi-convex functions on metric measure spaces - Existence, uniqueness and Lipschitz continuouity, http://arxiv.org/abs/1410.3966.
  • [26] Karl-Theodor Sturm, Ricci Tensor for Diffusion Operators and Curvature-Dimension Inequalities under Conformal Transformations and Time Changes, http://arxiv.org/abs/1401.0687.
  • [27] , Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), no. 3, 273–297.
  • [28] Guofang Wang and Chao Xia, A sharp lower bound for the first eigenvalue on Finsler manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 6, 983–996.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0016
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