# Nonlinear spectrums of Finsler manifolds

## TL;DR

This paper explores the spectral properties of Finsler manifolds by introducing faithful dimension pairs to define and analyze the spectrum of the nonlinear Finsler-Laplacian, extending classical spectral bounds and applications.

## Contribution

It introduces the concept of faithful dimension pairs for Finsler spectra and constructs several based on topological invariants, extending spectral bounds and linking to Bakry-Émery spectra.

## Key findings

- Established bounds for eigenvalues of Finsler-Laplacian
- Constructed faithful dimension pairs using topological invariants
- Linked Finsler spectral theory to Bakry-Émery spectrum

## 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-\'Emery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik-Schnirelmann dimension pair.