On the Genus and Area of Constant Mean Curvature Surfaces with Bounded Index

Compact Constant Mean Curvature Surfaces with Low Genus

We describe numerical experiments that suggest the existence of compact constant mean curvature surfaces. Our surfaces come in three dihedrally symmetric families with the genus ranging from 3 to 5, 7 to 10, and 3 to 9, respectively; there are further surfaces with the symmetry of the Platonic polyhedra and genera 6, 12, and 30. We use the algorithm of Oberknapp and the second author that defin...

Surfaces of Constant Mean Curvature Bounded by Convex Curves

This paper proves that an embedded compact surface in the Euclidean space with constant mean curvature H 6= 0 bounded by a circle of radius 1 and included in a slab of width 1=jHj is a spherical cap. Also, we give partial answers to the problem when a surface with constant mean curvature and planar boundary lies in one of the halfspaces determined by the plane containing the boundary, exactly, ...

Constant Mean Curvature Surfaces of Any Positive Genus

We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus g ≥ 1, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.

Discrete Constant Mean Curvature Surfaces and Their Index

We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as disc...

Surfaces of Constant Mean Curvature