A Heat Flow for Diffeomorphisms of Flat Tori

Abstract

In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} \Delta u$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the induced metric to establish two-sided bounds on the singular values of Du, which shows that unlike harmonic map heat flow, this flow preserves diffeomorphisms. A change of variables for Du then allows us to establish a $C^\alpha$ estimate for the coefficient of the tension field, and thus (thanks to the quasilinear structure and the Schauder estimates) we get full regularity and long-time existence. We conclude with some energy estimates to show convergence to an affine diffeomorphism.