Embedded Minimal Surfaces in Three Manifolds

Johns Hopkins University, Baltimore MD

Investigators

Abstract

We propose various directions for studying embedded minimal surfaces in three-manifolds. This includes the study of the convergence of such surfaces, how they degenerate and whether or not their Morse index is bounded. There are many possible applications of results along these lines including to the spherical space-form problem, the topology of three-manifolds with positive scalar curvature etc. In addition, these results should yield new insight into minimal surfaces in Euclidean space. This project focuses on minimal surfaces in three-dimensional manifolds. Minimal surfaces are critical points for area (e.g., soap films are least area surfaces) and arise naturally in many problems in mathematics and in the other sciences. Many of the classical results in this area are on least area surfaces. In contrast, our main interest is in high index (i.e., highly unstable) embedded minimal surfaces with bounded topology - a well-known example is given by the standard helicoid. Roughly speaking, the point of some of our theorems is that the behavior of the helicoid is typical.