Geometry of constant mean curvature surfaces in ${\mathbb{R}}^3$

Abstract

The crowning achievement of this paper is the proof that round spheres are the only complete, simply-connected surfaces embedded in ${\mathbb{R}}^3$ with nonzero constant mean curvature. Fundamental to this proof are new results including the existence of intrinsic curvature and radius estimates for compact disks embedded in ${\mathbb{R}}^3$ with nonzero constant mean curvature. We also prove curvature estimates for compact annuli embedded in ${\mathbb{R}}^3$ with nonzero constant mean curvature and apply them to obtain deep results on the global geometry of complete surfaces of finite topology embedded in ${\mathbb{R}}^3$ with constant mean curvature.