Dimension-like functions and spectrums of Finsler manifolds

TL;DR

This paper investigates the spectral properties of compact Finsler manifolds, defining a spectrum based on Sobolev space sets and a dimension-like function, establishing existence of eigenfunctions, upper bounds, and exploring examples.

Contribution

It introduces a novel spectral framework for Finsler manifolds using dimension-like functions and constructs new examples to illustrate the theory.

Findings

Eigenfunctions always exist for each eigenvalue.

A Cheng type upper bound estimate for eigenvalues is established.

Constructed and analyzed interesting examples of Finsler manifolds.

Abstract

In this paper, we study the spectral problem on a compact Finsler manifold with or without boundary. More precisely, given a certain collection of sets in Sobolev space $H^{1,2}(M)$ and a dimension-like function, we can define a corresponding spectrum. Such a spectrum satisfies nice properties. In particular, the eigenfunction corresponding to each eigenvalue always exists. And a Cheng type upper bound estimate for eigenvalues is obtained. Moreover, some interesting examples are constructed and investigated in this paper.