Sigma Models with Repulsive Potentials

TL;DR

This paper develops new techniques for geometric PDEs, improving regularity results and analyzing energy minimizing maps with singular potentials into Hermitian matrices related to Grassmannians.

Contribution

It introduces novel methods for monotonicity relations in geometric PDEs and applies them to regularity and energy minimization problems involving singular potentials.

Findings

Enhanced epsilon regularity results for harmonic maps

Effective analysis of energy minimizing maps with singular potentials

New techniques applicable to geometric PDEs involving Hermitian matrices

Abstract

Motivated by questions arising in the study of harmonic maps and Yang Mills theory, we study new techniques for producing optimal monotonicity relations for geometric partial differential equations. We apply these results to sharpen epsilon regularity results. As a sample application, we analyze energy minimizing maps from compact manifolds to the space of Hermitian matrices, where the energy of the map includes the usual kinetic term and a singular potential designed to force the image of the map to lie in a set homotopic to a Grassmannian.