Gluing and moduli for noncompact geometric problems

In this paper we survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are: I. Complete properly immersed minimal surfaces in $\RR^3$ with finite total curvature. II. Complete embedded surfaces of constant mean curvature in $\RR^3$ with finite topology. III. Complete conformal metrics of constant positive scalar curvature on $M^n \setminus \Lambda$, where $M^n$ is a compact Riemannian manifold, $n\geq3$ and $\Lam \subset M$ is closed. The existence results we discuss for each of these problems are ones whereby known solutions (sometimes satisfying certain nondegeneracy hypotheses) are glued together to produce new solutions. Although this sort of procedure is quite well-known, there have been some recent advances on which we wish to report here. We also discuss what has been established about the moduli spaces of all solutions to these problems, and report on some work in progress concerning global aspects of these moduli spaces. In the final section we present a new compactness result for the `unmarked moduli spaces' for problem III.