Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds

TL;DR

This paper establishes an $L^{d/2}$ energy gap for Yang-Mills connections on arbitrary closed Riemannian manifolds, extending previous results by removing curvature positivity constraints using advanced gradient inequalities.

Contribution

It proves a generalized $L^{d/2}$ energy gap for Yang-Mills connections on any closed Riemannian manifold, removing previous curvature restrictions.

Findings

Proves an $L^{d/2}$ energy gap for Yang-Mills connections on arbitrary manifolds.

Removes curvature positivity constraints from previous results.

Utilizes the Lojasiewicz-Simon gradient inequality for the proof.

Abstract

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to remove a positivity constraint on a combination of the Ricci and Riemannian curvatures in a previous $L^{d/2}$-energy gap result due to Gerhardt (2010) and a previous $L^\infty$-energy gap result due to Bourguignon, Lawson, and Simons (1981, 1979), as well as an $L^2$-energy gap result due to Nakajima (1987) for a Yang-Mills connection over the sphere, $S^d$, but with an arbitrary Riemannian metric. The main correction in this version involves replacement of the role of Corollary 4.3 due to Uhlenbeck (1985) and Theorem 5.1 due to the author in the published version of this article at http://dx.doi.org/10.1016/j.aim.2017.03.023 by that of Theorems 1 and 9 due to the author in arXiv:1906.03954.