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 twosided 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 longtime existence. We conclude with some energy estimates to show convergence to an affine diffeomorphism.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.08317
 Bibcode:
 2016arXiv160908317A
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 53C44;
 35K45;
 35K59;
 58J35
 EPrint:
 10 pages