Convergence of the Yang-Mills flow on ALE gravitational instantons

TL;DR

This paper proves a convergence theorem for the Yang-Mills flow on certain noncompact 4-manifolds with hyperK"ahler ALE geometry, extending previous results to a noncompact setting.

Contribution

It establishes a noncompact version of the parabolic gap theorem for the Yang-Mills flow on ALE gravitational instantons.

Findings

Proves sharp convergence of the Yang-Mills flow on ALE hyperK"ahler 4-manifolds.

Extends the parabolic gap theorem to noncompact settings.

Provides new insights into gauge theory on noncompact manifolds.

Abstract

We prove a sharp convergence theorem for the Yang-Mills flow on an $\mathrm{S}\mathrm{U}(r)$-bundle over a locally hyperK\"ahler ALE 4-manifold. Our main result is a noncompact version of the "parabolic gap theorem" previously established by the authors.