Nonlinear spectrums of Finsler manifolds

Alexandru Kristály; Zhongmin Shen; Lixia Yuan; Wei Zhao

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arXiv · 1907.01182

Nonlinear spectrums of Finsler manifolds

Alexandru Kristály·Zhongmin Shen·Lixia Yuan·Wei Zhao

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Abstract

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler-Laplacian operator, we introduce \textit{faithful dimension pairs} by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik-Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry-Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik-Schnirelmann dimension pair.

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BibTeX RIS

Alexandru Kristály, Zhongmin Shen, Lixia Yuan, Wei Zhao. 2019-10-27. Nonlinear spectrums of Finsler manifolds. https://arxiv.org/abs/1907.01182

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