Constant Mean Curvature Surfaces with Cylindrical Ends

Abstract

Abstract. We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.Surfaces which minimize area under a volume constraint have constant mean curvature (cmc); this condition can be expressed as a nonlinear partial differential equation. We are interested in complete cmc surfaces properly embedded in R 3 ; we rescale them to have mean curvature one. For technical reasons, we consider a slight generalization of embeddedness (introduced by Alexandrov [2]): An immersed surface is almost embedded if it bounds a properly immersed three-manifold.Alexandrov [1,2] showed that the round sphere is the only compact almost embedded cmc surface. The next case to consider is that of finite-topology surfaces, homeomorphic to a compact surface with a finite number of points removed. A neighborhood of any of these punctures is called an end of the surface. The unduloids, cmc surfaces of revolution described by Delaunay [4], are genus-zero examples with two ends. Each is a solution of an ordinary differential equation; the entire family is parametrized by the unduloid necksize, which ranges from zero (at the singular chain of spheres) to π (at the cylinder).Over the past decade there has been increasing understanding of finite-topology almost embedded cmc surfaces. Each end of such a surface is asymptotic to an unduloid [12]. Meeks [16] showed there are no examples with a single end. The unduloids themselves are the only examples with two ends [12]. Kapouleas [10] has constructed examples (near the zero-necksize limit) with any genus and any number of ends greater than two.In this note we announce the classification of all almost embedded cmc surfaces with three ends and genus zero; we call these triunduloids (see figure). In light of the trousers decomposition for surfaces, triunduloids can be seen as the building blocks for more complicated almost embedded cmc surfaces [8]. Our main result determines explicitly the moduli space of triunduloids with labelled ends, up to Euclidean motions. Since triunduloids are transcendental objects, and are not described by any ordinary differential equation, it is remarkable to have such a complete and explicit determination for their moduli space.

The Geometry of Bubbles and Foams

“…The conservation of forces (the homology invariance of F ) means that these forces on the ends must sum to zero. This balancing condition is the essential ingredient in classifying such cmc surfaces, and combined with some spherical trigonometry can lead to a complete classification of the three-ended surfaces [13,14].…”

Coplanar constant mean curvature surfaces

“…As a consequence we have the following constraints on the necksizes of a coplanar k-unduloid of genus 0. (These do not hold for higher genus, as shown by the coplanar k-unduloids of genus 1 with all ends cylindrical [GKS3].) Figure 1.…”

Constant mean curvature surfaces with three ends

“…In this note we announce the classification of all almost embedded cmc surfaces with three ends and genus zero; we call these triunduloids (see figure). In light of the trousers decomposition for surfaces, triunduloids can be seen as the building blocks for more complicated almost embedded cmc surfaces [8]. Our main result determines explicitly the moduli space of triunduloids with labelled ends, up to Euclidean motions.…”