What are the shapes of embedded minimal surfaces and why?

Abstract

Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The study is divided into three cases, depending on the topology of the surface. Case one is where the surface is a disk, in case two the surface is a planar domain (genus zero), and the third case is that of finite (non-zero) genus. The complete understanding of the disk case is applied in both cases two and three. As we will see, the helicoid, which is a double spiral staircase, is the most important example of an embedded minimal disk. In fact, we will see that every such disk is either a graph of a function or part of a double spiral staircase. The helicoid was discovered to be a minimal surface by Meusnier in 1776. For planar domains the fundamental examples are the catenoid, also discovered by Meusnier in 1776, and the Riemann examples discovered by Riemann in the beginning of the 1860s. Finally, for general fixed genus an important example is the recent example by Hoffman-Weber-Wolf of a genus one helicoid. In the last section we discuss why embedded minimal surfaces are automatically proper. This was known as the Calabi-Yau conjectures for embedded surfaces. For immersed surfaces there are counter-examples by Jorge-Xavier and Nadirashvili.