Obata’s Rigidity Theorem for Metric Measure Spaces

Christian Ketterer

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 278-295, electronic only
  • ISSN: 2299-3274

Abstract

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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.

How to cite

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Christian Ketterer. "Obata’s Rigidity Theorem for Metric Measure Spaces." Analysis and Geometry in Metric Spaces 3.1 (2015): 278-295, electronic only. <http://eudml.org/doc/275923>.

@article{ChristianKetterer2015,
abstract = {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.},
author = {Christian Ketterer},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {eigenvalue; suspension; Hessian; convexity},
language = {eng},
number = {1},
pages = {278-295, electronic only},
title = {Obata’s Rigidity Theorem for Metric Measure Spaces},
url = {http://eudml.org/doc/275923},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Christian Ketterer
TI - Obata’s Rigidity Theorem for Metric Measure Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 278
EP - 295, electronic only
AB - 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.
LA - eng
KW - eigenvalue; suspension; Hessian; convexity
UR - http://eudml.org/doc/275923
ER -

References

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  1. [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  Zbl1307.49044
  2. [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.  Zbl1312.53056
  3. [3] , Metric measure spaces with Riemannian Ricci curvature bounded from below, DukeMath. J. 163 (2014), no. 7, 1405– 1490.  Zbl1304.35310
  4. [4] Anders Björn and Jana Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich, 2011.  Zbl1231.31001
  5. [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.  Zbl0980.58020
  6. [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.  Zbl1196.53027
  7. [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.  Zbl1327.53050
  8. [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  Zbl1329.53059
  9. [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.  Zbl1227.31001
  10. [10] Nicola Gigli, On the differential structure of metric measure spaces and applications, Providence, Rhode Island: American Mathematical Society, 2015.  Zbl1325.53054
  11. [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.  Zbl1283.31002
  12. [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. [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. [14] Yin Jiang and Hui-Chun Zhang, Sharp spectral gaps on metric measure spaces, http://arxiv.org/abs/1503.00203.  Zbl06560647
  15. [15] Christian Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J.Math. Pures Appl. (9) 103 (2015), no. 5, 1228–1275.  Zbl1317.53064
  16. [16] , Ricci curvature bounds for warped products, J. Funct. Anal. 265 (2013), no. 2, 266–299.  Zbl1284.53072
  17. [17] Pawel Kröger, On the spectral gap for compact manifolds, J. Differential Geom. 36 (1992), no. 2, 315–330.  
  18. [18] John Lott and Cédric Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (2007), no. 1, 311–333.  Zbl1119.53028
  19. [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] Zbl0115.39302
  20. [20] Shin-ichi Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), no. 4, 805–828.  Zbl1176.28016
  21. [21] , Products, cones, and suspensions of spaces with themeasure contraction property, J. Lond.Math. Soc. (2) 76 (2007), no. 1, 225–236.  
  22. [22] A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), no. 1, 123–148.  Zbl0903.53045
  23. [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.  Zbl1278.53077
  24. [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.  Zbl1275.49087
  25. [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. [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. [27] , Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), no. 3, 273–297.  Zbl0854.35016
  28. [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.  Zbl1286.35179

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