A Mean-Value Laplacian For Finsler Spaces

Abstract

On a Riemannian space, the Laplace operator (both for forms and functions) is a natural and important operator. It leads to the Hodge Decomposition Theorem, which gives topological information about the space, and is essential to investigating the diffusion of heat. These considerations also make sense on the more general Finsler spaces, but so far it is not clear what we should use as a Laplacian on Finsler spaces. In this paper, we seek to generalize the Laplacian (first for functions and then for forms) on a Riemannian space to a Laplacian on a Finsler space. We do this by generalizing an important property of the Laplacian on Riemannian space, and that is that the Laplacian (at least infinitesimally) measures the average value of a function around a point.

This paper contains results obtained for a doctoral dissertation, under the supervision of Prof. John Bland, at the University of Toronto, and was presented at the University of Alberta in August, 1997.