Doubling and Desingularization Constructions for Minimal Surfaces

Abstract

In the first part of the paper we discuss the current status of the application of the gluing methodology to doubling and desingularization constructions for minimal surfaces in Riemannian three-manifolds. In particular a doubling construction for equatorial spheres in $S^3(1)$ is announced. Aspects of the current understanding of existence and uniqueness questions for closed minimal embedded surfaces in $S^3(1)$ are also discussed, and some new uniqueness questions are proposed. In the second part of the paper we discuss some of the ideas and provide an outline for a general desingularization construction without imposed symmetries. This paper is the author's contribution to the volume in honor of Professor Richard M. Schoen's sixtieth birthday.