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

2017/06/28 by Feehan, Paul M. N.
#57R57 #58D27 #70S15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 58E15 #secondary 37D15

paper · doi:10.48550/arxiv.1706.09349

Abstract

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

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