Geometry of the ends of the moduli space of anti-self-dual connections

Abstract: Let $X$ be a closed, four-dimensional, oriented, smooth manifold with a Riemannian metric, $g$, let $G$ be a compact Lie group, and $P$ be a principal $G$ bundle over $X$. D. Groisser and T. Parker (1987, 1989) and S. K. Donaldson (1990) conjectured that the moduli space of $g$-anti-self-dual connections on $P$, endowed with the $L^2$ metric, has finite volume and diameter. The purpose of this article is to prove this conjecture under the following additional hypotheses. Suppose that $g$ is generic and $X$ is simply-connected. If (i) $G=SU(2)$ or $SO(3)$ and $b^+(X)=0$ or (ii) $G=SO(3)$ and $w_2(P)\neq 0$, where $w_2(P)$ is the second Stiefel-Whitney class of $P$, then we prove that the moduli space of $g$-anti-self-dual connections on $P$ has finite volume and diameter with respect to the $L^2$ metric. Our development of the bubble-tree compactification of the moduli space of $g$-anti-self-dual connections --- based on ideas of Sacks and Uhlenbeck for sequences of harmonic maps from the two-sphere (1981), Taubes (1988) for sequences of Yang-Mills connections, and Parker and Wolfson (1993, 1996) for sequences of pseudo-holomorphic maps --- provides one of the key technical tools used in the proof.