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

Abstract

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 $d\ge 2$. We apply our version of the Łojasiewicz–Simon gradient inequality [16], [19] 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 [23, Theorem 1.2] and a previous $L^{\infty }$-energy gap result due to Bourguignon, Lawson, and Simons [10, Theorem C], [11, Theorem 5.3], as well as an $L^2$-energy gap result due to Nakajima [42, Corollary 1.2] for a Yang–Mills connection over the sphere, $S^d$, but with an arbitrary Riemannian metric.