Totally Geodesic Boundaries of Yang-mills Moduli Spaces

Moduli spaces M of self-dual SU(2) connections (“instantons”) over a compact Riemannian 4.manifold (hl, g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson’s compactification; in fact the boundary of the completion is an isometric copy of (M, 4n2g) ([GP2]). In this paper we show that the boundary is, furthermore, a totally geodesic submanifold of the completion. Along the way, we prove a regularity theorem: the continuous extension of g to the “collar region” of M is C1,a (in the conventional scale/center coordinates) for small LY > 0. The proofs rely on some new weighted Sobolev inequalities for concentrated instantons, in which the only dependence of the Sobolev constants on the connection is through the concentration parameter X. The exponent in the weighting function translates into the Holder exponent in the regularity theorem.