GAUGE THEORY ON OPEN MANIFOLDS OF BOUNDED GEOMETRY

Abstract

On compact manifolds (Mn, g) and for r>n/2+1 the configuration space [Formula: see text] is a well-defined object. [Formula: see text] is an affine space with a Sobolev space as vector space, and [Formula: see text] a Hilbert Lie group which acts smoothly and properly on [Formula: see text]. [Formula: see text] is a stratified space with Hilbert manifolds as strata. The existence problem has been solved for many interesting cases by Cliff Taubes and the description of the moduli space of instantons has been given by Donaldson. On noncompact manifolds none of the approaches of the compact case is further valid. We present here an intrinsic, self-consistent approach for gauge theory on open manifolds of bounded geometry up to order n/2+2. The main idea is to endow the space CP of gauge potentials and the gauge group with an intrinsic Sobolev topology. Bounded geometry of the underlying manifold and the considered connections provides all the Sobolev theorems which are needed to prove the existence of instantons if G=SU(2). We prove the existence of instantons if (M4, g) satisfies a certain spectral condition and has a positive definite L2 intersection form.