Entropy, Stability, and Yang-Mills flow

TL;DR

This paper introduces an entropy concept for Yang-Mills connections, characterizes stability via spectral analysis, and establishes a gap theorem, advancing understanding of singularities in Yang-Mills flow.

Contribution

It defines a new entropy for Yang-Mills connections, characterizes entropy stability through spectral analysis, and proves a gap theorem for solitons.

Findings

Entropy is defined for Yang-Mills connections with solitons as critical points.

Entropy stability is characterized by the spectrum of a linear operator.

A gap theorem for solitons is established.

Abstract

Following work of Colding-Minicozzi, we define a notion of entropy for connections over $\mathbb R^n$ which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying "generic singularities" of Yang-Mills flow, and we discuss the differences in this strategy in dimension $n=4$ versus $n \geq 5$.