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

Abstract

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\ge 2$, 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 $d\ge 2$, 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.

44 pages. Relies on arXiv:1502.00668 by the author and on arXiv:1510.03817 by the author and Maridakis for background material and supporting results

References in corpus (5)

• 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
• Yang-Mills moduli spaces over an orientable closed surface via Fréchet reduction
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Cited by in corpus (4)

• Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds
• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
• Morse theory for the Yang-Mills energy function near flat connections
• On a topology property for the moduli space of Kapustin-Witten equations