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.