Heat Flow on Finsler Manifolds

This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, .) : T(x)M -> R(+) on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L(2) (M, m) for the energy epsilon(u) = 1/2 integral(M) F(2)(del u)dm as gradient flow on the reverse L(2)-Wasserstein space P(2)(M) of probability measures on M for the relative entropy Ent(u) = integral(M) u log u dm. Both approaches depend on the choice of a measure in on M and then lead to the same nonlinear evolution semigroup. We prove C(1,alpha) regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be C(2). Moreover, we derive pointwise comparison results a la Cheeger-Yau and integrated upper Gaussian estimates a la Davies. (C) 2008 Wiley Periodicals, Inc.