Edge and gluing theorems associated to the singular Yamabe problem

This dissertation examines the singular Yamabe problem in a variety of cases. First we prove that a certain class of singular partial differential equations (closely related to those that arise while studying the singular Yamabe problem) has solutions expressable as convergent sums of elementary functions, and parametrize such solutions. Secondly, we construct complete constant scalar curvature metrics on the complement of finite sets in various types of manifold. The technique used is a Cauchy data matching technique: We perturb metrics on the main manifold, and on well-known end manifolds, in such a way as to make them match up at a specified hypersurface in each manifold. We conclude that the two manifolds can be joined along this hypersurface. We also provide similar gluing arguments involving incomplete ends, and thus provide a link between the study of constant scalar curvature metrics which are complete on the complement of a point, and the study of constant scalar curvature metrics vanishing at the boundaries of small holes.