Entropy, stability and harmonic map flow

Abstract

Inspired by work of Colding-Minicozzi (2012) on mean curvature flow, Zhang (2012) 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 F \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 (2008) 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.