Embedded Minimal Surfaces in Three Manifolds

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