The rectified n-harmonic map flow with applications to homotopy classes

TL;DR

This paper introduces a new rectified n-harmonic map flow on closed Riemannian manifolds, proves its global existence with finite singularities, and applies it to energy minimization within homotopy classes.

Contribution

It develops a novel rectified n-harmonic map flow, proves its global existence and energy identity, and demonstrates applications to energy functional minimization in homotopy classes.

Findings

Global existence of the flow with finite singularities

Energy identity at singular times

Applications to minimizing energy functionals in homotopy classes

Abstract

We introduce a rectified $n$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed Riemannian manifold. We prove existence of a global solution, which is regular except for a finite number of points, of the rectified n-harmonic map flow and establish an energy identity for the flow at each singular time. Finally, we present two applications of the rectified n-harmonic map flow to minimizing the n-energy functional and the Dirichlet energy functional in a homotopy class.