We will 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. Some of our main results for minimal surfaces so far - the lamination theorem and the one-sided curvature estimate - are for properly embedded minimal disks. Recent years have seen breakthroughs on many long--standing problems in the theory of minimal surfaces, with important contributions from many mathematicians. The lamination theorem and the one-sided curvature estimate described here have played a key role and have been used by many people. Two of the important new directions are removing the assumption of properness and considering minimal surfaces with more general topological types. These results will have important implications.
The field of minimal surfaces dates back to early work of Euler in 1744 and Lagrange in 1762 and has remained a vibrant area of research for the last 250 years. Minimal surfaces appear frequently throughout science, dating back at least to the soap film experiments of the Belgian physicist Plateau in the first half of the nineteenth century. Their mathematical impact has been significant and has led to developments in geometry, topology, and partial differential equations. The subject has seen major developments recently, including answers to some long-standing open questions and a rather complete picture for properly embedded minimal disks (such as the helicoid which was originally discovered in 1776). However, much less is known when these assumptions are removed; understanding this is a key part of our research and the answers are likely to lead to further developments.
