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.
