Critical-exponent Sobolev norms and the slice theorem for the quotient space of connections

TL;DR

This paper develops optimal Sobolev norm estimates for connections on four-manifolds, enabling a precise Coulomb gauge slice theorem for the quotient space of connections, with potential applications to non-linear PDEs and gauge theory.

Contribution

It introduces a collection of critical-exponent Sobolev norms and Green's operator estimates that lead to an optimal slice theorem for the space of connections, extending to degenerating reference connections.

Findings

Established Sobolev norm estimates depending only on eigenvalues and curvature norms.

Proved an optimal Coulomb gauge slice theorem for the quotient space of connections.

Applicable to manifolds of arbitrary dimension with degenerating reference connections.

Abstract

The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green's operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature of the connection. Following Taubes, we describe a collection of critical-exponent Sobolev norms and general Green's operator estimates depending only on first positive eigenvalues or the L^2 norm of the connection's curvature. Such estimates are particularly useful in the gluing construction of solutions to non-linear partial differential equations depending on a degenerating parameter, such as the approximate, reference solution in the anti-self-dual or PU(2) monopole equations. We apply them here to prove an optimal slice theorem for the quotient space of connections. The result is optimal in the sense that if a point [A] in the quotient space is known to be just L^2_1-close enough to a reference point [A_0], then the connection A can be placed in Coulomb gauge relative to the connection A_0, with all constants depending at most on the first positive eigenvalue of the covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In this paper we shall for simplicity only consider connections over four-dimensional manifolds, but the methods and results can adapted to manifolds of arbitrary dimension to prove slice theorems which apply when the reference connection is allowed to degenerate.