Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions

TL;DR

This paper establishes optimal Lojasiewicz-Simon inequalities for Morse-Bott functions related to Yang-Mills energy, extending Uhlenbeck's flat connection results and providing new proofs for inequalities near flat and irreducible connections.

Contribution

It generalizes Lojasiewicz-Simon inequalities to Morse-Bott Yang-Mills functions and proves the Morse-Bott property for certain Yang-Mills connections, with applications to gauge theory.

Findings

Proved optimal Lojasiewicz-Simon inequalities for Yang-Mills energy functions.

Established Morse-Bott property for irreducible Yang-Mills connections on Riemann surfaces.

Extended Uhlenbeck's flat connection existence result to $L^{d/2}$-small curvature case.

Abstract

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with $L^{d/2}$-small curvature. We prove an optimal Lojasiewicz-Simon gradient inequality for abstract Morse-Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis in arXiv:1510.03817. We apply this result to prove the optimal Lojasiewicz-Simon gradient inequality for the self-dual Yang-Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang-Mills energy function over closed Riemannian manifolds of dimension $d \geq 2$, when known to be Morse-Bott at a given Yang-Mills connection. We also prove the optimal Lojasiewicz-Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map. We also prove the Morse-Bott property for irreducible Yang-Mills $U(n)$ connections over Riemann surfaces and hence a new proof of the optimal Lojasiewicz-Simon gradient inequality for such critical points.