2-Type Surfaces of Constant Curvature in $S^n$

Surfaces of Constant Mean Curvature

Bonnesen-type inequalities for surfaces of constant curvature

A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant Gauss curvature κ > 0) and the hyperbolic plane (having constant Gauss curvature κ < 0). These generalized inequalities each converge to the clas...

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...

Willmore Surfaces of Constant Möbius Curvature

We study Willmore surfaces of constant Möbius curvature K in S. It is proved that such a surface in S must be part of a minimal surface in R or the Clifford torus. Another result in this paper is that an isotropic surface (hence also Willmore) in S of constant K could only be part of a complex curve in C ∼= R or the Veronese 2-sphere in S. It is conjectured that they are the only examples possi...