Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow

TL;DR

This paper establishes conditions for the global existence and convergence of solutions to gradient flows, particularly applying these results to the Yang-Mills gradient flow on various manifolds, including new convergence criteria near local minima.

Contribution

It proves a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional and applies it to demonstrate global existence and convergence of the Yang-Mills gradient flow under new conditions.

Findings

Flow exists globally near local minima in certain norms.

Flow converges to Yang-Mills connections under specified conditions.

Bubble singularities may occur in some cases, but solutions still exist for all time.

Abstract

In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riemannian manifolds of arbitrary dimension and apply the resulting framework to prove new results for the gradient flow equation for the Yang-Mills energy functional on a principal bundle, with compact Lie structure group, over a closed, smooth Riemannian manifolds, including the following. If the initial connection is close enough to a local minimum of the Yang-Mills energy functional, in a norm sense when the base manifold has arbitrary dimension or in an energy sense when the base manifold has dimension four, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection. If the initial connection is allowed to have arbitrary energy but we restrict to the setting of a Hermitian vector bundle over a compact, complex, Hermitian (but not necessarily Kaehler) surface and the initial connection has curvature of type (1,1), then the Yang-Mills gradient flow exists for all time, though bubble singularities may (and in certain cases must) occur in the limit as time tends to infinity.