Abstract

Following Taubes, we describe a collection of critical-exponent Sobolev norms, discuss their embedding and multiplication properties, and describe optimal Green’s operator estimates where the constants depend at most on the first positive eigenvalue of the covariant Laplacian of a G connection and the L² norm of the connection’s curvature, for arbitrary compact Lie groups G. Using these critical-exponent norms, we prove a sharp, global analogue of Uhlenbeck’s Coulomb gauge-fixing theorem, where the usual product connection over a ball is replaced by an arbitrary reference connection over the entire manifold. We also prove a quantitative version of the conventional slice theorem for the quotient space of G connections, with an invariant and sharp characterization of those points in the quotient space which are contained in the image of an L⁴ ball in the Coulomb-gauge slice. Our gauge-fixing and slice theorems use L₁² distance functions on the quotient space and the estimate constants depend at most on the first positive eigenvalue of the covariant Laplacian of the reference connection and the L² norm of its curvature.

Authors