Stability of surfaces with constant mean curvature

Stability and Bifurcation for Surfaces with Constant Mean Curvature

We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in R, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids inR.

Surfaces of Constant Mean Curvature

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...

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...

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...