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

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 $d\ge 2$. 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.

26 pages, incorporating final galley proof corrections and corrections to errors noticed since publication online at http://dx.doi.org/10.1016/j.aim.2017.03.023. Background and analytical results provided by arXiv:1409.1525, arXiv:1706.09349, and arXiv:1906.03954

References in corpus (6)

• Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds
• Energy gap for Yang-Mills connections, I: Four-dimensional closed Riemannian manifolds
• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• A proof on energy gap for Yang-Mills connection
• Morse theory for the Yang-Mills energy function near flat connections
• Some Energy Properties of Yang-Mills Connections

Cited by in corpus (9)

• Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds
• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
• Energy gap for Yang-Mills connections, I: Four-dimensional closed Riemannian manifolds
• Discreteness for energies of Yang-Mills connections over four-dimensional manifolds
• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• A proof on energy gap for Yang-Mills connection
• An Energy Gap for Complex Yang-Mills Equations
• Morse theory for the Yang-Mills energy function near flat connections
• Some boundedness properties of solutions to the complex Yang-Mills equations on closed $4$-manifolds