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

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.