Heat flow on Finsler manifolds

Abstract

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space ${\mathcal{P}}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on $M$ and then lead to the same nonlinear evolution semigroup. We prove ${\mathcal{C}}^{1,\alpha }$-regularity for solutions to the (nonlinear) heat equation on the Finsler space $(M,F,m)$. Typically, solutions to the heat equation will not be ${\mathcal{C}}^2$. Moreover, we derive pointwise comparison results a la Cheeger-Yau and integrated upper Gaussian estimates a la Davies.

arXiv comment36 pages, v4: minor corrections in Lemma 3.8, Theorem 3.9 (figures are not included from technical reasons, see the published one or the authors' websites for them)