Some results on the heat kernel asymptotics of the Laplace operator on Finsler spaces

Abstract

In this paper we consider Bao-Lackey's extension of the Laplace operator on a Finsler space. We prove that this operator is of Laplace type on scalars and on top degree forms, and compute the first heat coefficients. In exchange, the BL Laplacian on 1-forms is nonminimal and a study of its heat kernel asymptotics is more difficult. The results obtained in this paper for the 1-formed Laplacian concern Finsler surfaces and direct products of Finsler surfaces. We apply our computation of the heat coefficients to prove that, on Randers spaces, the scalar BL Laplacian and the scalar Laplacian of the metric $a_{ij}$ have the same eigenvalues if and only if the Randers space is Riemann.