Optimal Lojasiewicz-Simon Inequalities and Morse-Bott Yang-Mills Energy Functions

Feehan, Paul M. N. (2017) Optimal Lojasiewicz-Simon Inequalities and Morse-Bott Yang-Mills Energy Functions. (Preprint)

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Abstract

For any compact Lie group G, we prove that the Yang–Mills energy function obeys an optimal gradient inequality of Łojasiewicz–Simon type (exponent 1/2) near the critical set of flat connections on a principal G-bundle over a closed Riemannian manifold of dimension d ≥ 2 and so its gradient flow converges at an exponential rate to that critical set. We establish this optimal Łojasiewicz–Simon gradient inequality by three different methods. Our first proof gives the most general result by direct analysis and relies on our extension of a theorem due to Uhlenbeck [86] that gives existence of a flat connection on a principal G-bundle supporting a connection with L^d/2 -small curvature, existence of a Coulomb gauge transformation, and W^1,p Sobolev distance estimates for p > 1. Our second proof proceeds by first establishing an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis [31, Theorem 4]. Our third proof establishes the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map. We prove similar results 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.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 13:49
Last Modified: 12 Jul 2018 10:10
URI: http://dair.dias.ie/id/eprint/551

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