Gluing and Moduli for Noncompact Geometric Problems

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. Although the analysis underlying each of these problems differs somewhat from one case to the next, there are many common themes. The basic point which makes each of these problems tractable is the fact that the geometry of the ends of the manifolds of interest is well-understood. The ends of a surface Σ in case I are either asymptotically planes or catenoids, in case II the ends are asymptotically periodic, and in case III,