On the Sacks-Uhlenbeck flow of Riemannian surfaces

TL;DR

This paper investigates an alpha-flow related to the Sack-Uhlenbeck functional on Riemannian surfaces, demonstrating its convergence to harmonic maps and providing a simplified proof of energy identities in homotopy classes.

Contribution

It introduces an alpha-flow approach for the Sack-Uhlenbeck functional and proves its convergence to harmonic maps, offering a new, simplified proof of energy identities.

Findings

The alpha-flow converges to weak harmonic maps.

A simplified proof of energy identity for minimizing sequences.

The approach applies to Riemannian surfaces.

Abstract

In this paper, we study an $\alpha$-flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the $\alpha$-flows is a weak solution to the harmonic map flow. By an application of the $\alpha$-flow, we present a simple proof of an energy identity of a minimizing sequence in each homotopy class.