Abstract
We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature, if its first closed (resp. Neumann) eigenvalue of Finsler-Laplacian attains the sharp lower bound, then M is isometric to a circle (resp. a segment). Moreover, a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated. These generalize the corresponding results in recent literature.
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References
Bao D W, Chern S S, Shen Z M. An Introduction to Riemann-Finsler Geometry. New York: Springer-Verlag, 2000
Chen B. Some geometric and analysis problems in Finsler geometry. Report of postdoctoral Research, Zhejiang University, 2010, 4
Ge Y, Shen Z M. Eigenvalues and eigenfunctions of metric measure manifolds. Proc London Math Soc, 2001, 82: 725–746
Hang F B, Wang X D. A remark on Zhong-Yang’s eigenvalue estimate. Int Math Res Not. 2007, 18: Art ID rnm064, 9pp
Li P, Yau S T. Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace Operator, Proc Sympos Pure Math, XXXVI. Providence, RI: Amer Math Soc, 1980, 205–239
Lichnerowicz A. Geometrie des groupes de transforamtions. In: Travaux et Recherches Mathemtiques, III. Paris: Dunod, 1958
Obata M. Certain conditions for a Riemannian manifold tobe isometric with a sphere. J Math Soc Japan, 1962, 14: 333–340
Ohta S. Finsler interpolation inequalities. Calc Var Partial Differential Equations, 2009, 36: 211–249
Ohta S, Sturm K T. Bochner-Weitzenbock formula and Li-Yau estimates on Finsler manifolds. ArXiv:1105.0983
Shen Z M. Lectures on Finsler Geometry. Singapore: World Scientific Publishing, 2001
Sturm K T. On the geometry of metric measure spaces, I. Acta Math, 2006, 196: 65–131
Wang G F, Xia C. A sharp lower bound for the first eigenvalue on Finsler manifolds. ArXiv:1112.4401v1
Wang G F, Xia C. An optimal anisotropic poincare inequality for convex domains. ArXiv:1112.4398v1
Wu B Y, Xin Y L. Comparison theorems in Finsler geometry and their applications. Math Ann, 2007, 337: 177–196
Yin S T, He Q, Shen Y B. On lower bounds of the first eigenvalue of Finsler-Laplacian. ArXiv:1210.7606v1
Zhong J Q, Yang H C. On the estimates of the first eigenvalue of a compact Riemannian manifold. Sci China Ser A, 1984, 27: 1265–1273
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Yin, S., He, Q. & Shen, Y. On the first eigenvalue of Finsler manifolds with nonnegative weighted Ricci curvature. Sci. China Math. 57, 1057–1070 (2014). https://doi.org/10.1007/s11425-013-4707-9
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DOI: https://doi.org/10.1007/s11425-013-4707-9