Geometric Analysis; Minimal Surfaces, Geometric Flows, and Function Theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Recent years have seen breakthroughs on many long-standing problems in the theory of minimal surfaces, with important contributions from many mathematicians. A minimal surface is a natural generalization of geodesics. It is a surface that locally minimize area. The lamination theorem, the one-sided curvature estimate, the resolution of the Calabi-Yau conjectures for embedded surfaces, and the structure results for fixed genus surfaces (all joint work of the PI and Bill Minicozzi) have played a key role in recent breakthroughs and have been used by many people.The PI proposes, jointly with Bill Minicozzi, to continue our investigations on minimal surfaces and related areas of geometric analysis, including geometric evolution equations such as the mean curvature and Ricci flow, and on function theory. We also propose a joint project with Bruce Kleiner about mean curvature flow of mean convex sets and a joint project with Nancy Hingston about Morse index bounds of geodesics. Finally, we propose a project with Camillo De Lellis about Min-Max constructions and applications and a joint project with Camillo De Lellis and Bill Minicozzi on generic metrics.