Coplanar constant mean curvature surfaces

Coplanar Constant Mean Curvature Surfaces

We consider constant mean curvature surfaces with finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane...

New Constant Mean Curvature Surfaces

We use the DPW construction [5] to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the t...

Surfaces of Constant Mean Curvature

Constant mean curvature surfaces with cylindrical ends

R. Schoen has asked whether the sphere and the cylinder are the only complete (almost) embedded constant mean curvature surfaces with finite absolute total curvature. We propose an infinite family of such surfaces. The existence of examples of this kind is supported by results of computer experiments we carried out using an algorithm developed by Oberknapp and Polthier. The cylinder of radius 1...

Surfaces with maximal constant mean curvature

In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This bound allows us to find a maximizer for the constant mean curvature of a surface ...