Entropy, stability, and harmonic map flow

TL;DR

This paper explores the concept of entropy stability in harmonic map flow, establishing its equivalence with $$-stability, analyzing instability of high-entropy solitons, and examining long-term behavior in convex targets.

Contribution

It proves the equivalence of entropy stability with $$-stability and investigates stability and convergence properties of harmonic map flow into convex domains.

Findings

Entropy stability is equivalent to $$-stability.

High-entropy solitons are generally unstable.

Long-time existence and convergence are observed in convex target domains.

Abstract

Inspired by work of Colding-Minicozzi on mean curvature flow, Zhang introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable $\mathcal F$-stability. Then, focusing on the case of spherical targets, we prove a general instability result for high-entropy solitons. Finally, we exploit results of Lin-Wang to observe long time existence and convergence results for maps into certain convex domains and how they relate to generic singularities of harmonic map flow.