Parabolic gap theorems for the Yang-Mills energy

2024/12/30 by Anuk Dayaprema, Dayaprema, Anuk, Alex Waldron +1
Mathematics · Physics and Astronomy · #Spectral Theory in Mathematical Physics #Black Holes and Theoretical Physics

paper · pdf · doi:10.48550/arxiv.2412.21050

Abstract

We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an SU(r)-bundle of charge κ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than 4 π² ( |κ| + 2 ) deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-Kähler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose \mathfraksp(1) curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive.

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