Frontiers of Mathematics in China >
Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below
Received date: 06 Feb 2017
Accepted date: 13 Feb 2018
Published date: 28 Mar 2018
Copyright
We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.
Songting YIN . Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below[J]. Frontiers of Mathematics in China, 2018 , 13(2) : 435 -448 . DOI: 10.1007/s11464-018-0692-1
| 1 |
Calabi E. On manifolds with nonnegative Ricci curvature II. Notices Amer Math Soc, 1975, 22: A-205
|
| 2 |
Chavel I. Riemannian Geometry: A Modern Introduction. New York: Cambridge Univ Press, 1993
|
| 3 |
Lott J. Some geometric properties of the Bakry-Émery-Ricci tensor. Comment Math Helv, 2003, 78(4): 865–883
|
| 4 |
Ohta S. Finsler interpolation inequalities. Calc Var Partial Differential Equations, 2009, 36: 211–249
|
| 5 |
Ohta S, Sturm K T. Heat flow on Finsler manifolds. Comm Pure Appl Math, 2009, 62: 1386–1433
|
| 6 |
Ohta S, Sturm K T. Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds. Adv Math, 2014, 252: 429–448
|
| 7 |
Qian Z M. Estimates for weighted volumes and applications. Q J Math, Oxford Ser (2), 1997, 48(190): 235–242
|
| 8 |
Shen Z M. Volume comparison and its applications in Riemann-Finsler geometry. Adv Math, 1997, 128: 306–328
|
| 9 |
Shen Z M. Lectures on Finsler Geometry. Singapore: World Sci, 2001
|
| 10 |
Sturm K T. On the geometry of metric measure spaces I. Acta Math, 2006, 196: 65–131
|
| 11 |
Wei G F, Wylie W. Comparison geometry for the Bakry-Émery Ricci tensor. J Differential Geom, 2009, 83: 377–405
|
| 12 |
Wu B Y. Volume form and its applications in Finsler geometry. Publ Math Debrecen, 2011, 78(3-4): 723–741
|
| 13 |
Wu B Y, Xin Y L. Comparison theorems in Finsler geometry and their applications. Math Ann, 2007, 337: 177–196
|
| 14 |
Yau S T. Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ Math J, 1976, 25: 659–670
|
| 15 |
Yin S T, He Q, Zheng D X. Some comparison theorems and their applications in Finsler geometry. J Inequal Appl, 2014, 2014: 107
|
| 16 |
Zhao W, Shen Y B. A universal volume comparison theorem for Finsler manifolds and related results. Canad J Math, 2013, 65: 1401–1435
|
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