Stabilities of homothetically shrinking Yang-Mills solitons

TL;DR

This paper introduces new stability concepts for shrinking Yang-Mills solitons, linking entropy and second variation, and explores their implications for singularity formation and classification in Yang-Mills flow.

Contribution

It defines entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons and establishes their relationship, with applications to singularity analysis and classification.

Findings

Entropy-stability implies F-stability for non-descending solitons.

Yang-Mills flow in four dimensions cannot form Type-I singularities.

A gap theorem for homothetically shrinking solitons is proved.

Abstract

In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity; and we obtain a gap theorem for homothetically shrinking solitons.