Eigenvalues control for a Finsler--Laplace operator

TL;DR

This paper demonstrates that the spectrum of a Finsler--Laplace operator can be controlled by bi-Lipschitz equivalence of Finsler metrics and explores spectral bounds on Finsler surfaces, including examples of highly non-reversible metrics with large eigenvalues.

Contribution

It extends spectral control results from Riemannian to Finsler geometry and provides new examples of metrics with large eigenvalues on surfaces.

Findings

Spectrum is controlled by bi-Lipschitz equivalence of Finsler metrics.

Spectral bounds depend on surface topology and quasireversibility constant.

Existence of highly non-reversible metrics with arbitrarily large first eigenvalue.

Abstract

Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue.