A natural Finsler--Laplace operator

TL;DR

This paper introduces a novel Finsler-Laplace operator defined via an averaging process that leverages a dynamical approach, enabling explicit spectral computations for certain Finsler metrics.

Contribution

It proposes a new, connection-free definition of the Finsler-Laplace operator using dynamical methods and demonstrates explicit spectral calculations for specific metrics.

Findings

The new operator is defined without connections or local coordinates.

Explicit spectral data are computed for Katok-Ziller metrics.

The approach broadens the understanding of spectral geometry in Finsler settings.

Abstract

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representations and computations of spectral data.