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Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator.(English)Zbl 0870.60071

The classical theorem of Myers on diameter \([M]\) states that if \((M,g)\) is a complete, connected Riemannian manifold of dimension \(n\) \((\geq 2)\) such that \(\text{Ric}\geq(n-1)g\), then its diameter \(D=D(M)\) is less than or equal to \(\pi\). The authors prove an analogue of Myers’s theorem for an abstract Markov generator and provide at the same time a new analytic proof of this result based on Sobolev inequalities. In particular, how to get exact bounds on the diameter in terms of the Sobolev constant is shown.

MSC:

60J35 Transition functions, generators and resolvents
47D07 Markov semigroups and applications to diffusion processes

Cite

References:

[1]S. Aida, T. Masuda, and I. Shigekawa, Logarithmic Sobolev inequalities and exponential integrability , J. Funct. Anal. 126 (1994), no. 1, 83-101. ·Zbl 0846.46020 ·doi:10.1006/jfan.1994.1142
[2]Th. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. ·Zbl 0512.53044
[3]D. Bakry, Inégalités de Sobolev faibles: un critère \(\Gamma_ 2\) , Séminaire de Probabilités, XXV, Lecture Notes in Math., vol. 1485, Springer, Berlin, 1991, pp. 234-261. ·Zbl 0745.60084
[4]D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes , Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 1-114. ·Zbl 0856.47026
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[6]W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality , Ann. of Math. (2) 138 (1993), no. 1, 213-242. JSTOR: ·Zbl 0826.58042 ·doi:10.2307/2946638
[7]I. Chavel, Riemannian geometry-a modern introduction , Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. ·Zbl 0810.53001
[8]J. Cheeger, The relation between the Laplacian and the diameter for manifolds of non-negative curvature , Arch. Math. (Basel) 19 (1968), 558-560. ·Zbl 0177.50201 ·doi:10.1007/BF01898781
[9]S.-Y. Cheng, Eigenvalue comparison theorems and its geometric applications , Math. Z. 143 (1975), no. 3, 289-297. ·Zbl 0329.53035 ·doi:10.1007/BF01214381
[10]E. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères , to appear in Bull. Sci. Math. (2).
[11]S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry , Universitext, Springer-Verlag, Berlin, 1990. ·Zbl 0716.53001
[12]S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes , Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 151-165. ·Zbl 0528.53040 ·doi:10.5802/aif.921
[13]M. Ledoux, Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter , J. Math. Kyoto Univ. 35 (1995), no. 2, 211-220. ·Zbl 0836.60074
[14]S. B. Myers, Connections between differential geometry and topology , Duke Math. J. 1 (1935), 376-391. ·Zbl 0012.27502 ·doi:10.1215/S0012-7094-35-00126-0
[15]M. Obata, Certain conditions for a Riemannian manifold to be iosometric with a sphere , J. Math. Soc. Japan 14 (1962), 333-340. ·Zbl 0115.39302 ·doi:10.2969/jmsj/01430333
[16]O. S. Rothaus, Hypercontractivity and the Bakry-Emery criterion for compact Lie groups , J. Funct. Anal. 65 (1986), no. 3, 358-367. ·Zbl 0589.58036 ·doi:10.1016/0022-1236(86)90025-X
[17]V. A. Toponogov, Riemann spaces with curvature bounded below , Uspehi Mat. Nauk 14 (1959), no. 1 (85), 87-130. ·Zbl 0114.37504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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