Lojasiewicz--Simon gradient inequalities for the harmonic map energy function

TL;DR

This paper establishes Lojasiewicz--Simon gradient inequalities for the harmonic map energy function on Riemannian manifolds, extending previous results with minimal regularity assumptions using Sobolev spaces.

Contribution

It applies abstract gradient inequalities to harmonic map energy, generalizing earlier inequalities and reducing regularity requirements on maps between manifolds.

Findings

Generalized Lojasiewicz--Simon inequalities for harmonic maps

Extended applicability with minimal regularity assumptions

Unified framework building on previous inequalities

Abstract

We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient inequalities for the harmonic map energy function generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997).