| This thesis consists of two constructions of complete minimal surfaces as limits of compact minimal surfaces.;In the first part, we construct the Riemann examples as the limit of compact minimal annuli. The Colding-Minicozzi structure theorems require that singular limits have very specific structure; we prove smooth convergence by showing that these structures cannot be the limit of our sequence of minimal annuli. In order to control the topology of the limit, we also require an upper bound on the minimum length of closed geodesics on our surfaces.;In general, the only two types of singular laminations that can occur as limits of closed embedded minimal surfaces in a 3-manifold of positive scalar curvature are accumulations of catenoids and non-proper helicoid-like limits. We provide an example of the second type in the second part of this thesis, where we show that there exists a metric with positive scalar curvature on S2 x S1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. Prior to the construction given here, no non-proper helicoid-like limits were known to exist as limits of closed surfaces.;These constructions are inspired by a recent example by D. Hoffman and B. White. |
