Constant mean curvature surfaces with three ends

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 that 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 ‫ޒ‬ 3 ; we rescale them to have mean curvature one. For technical reasons, we consider a slight generalization of embeddedness [introduced by Alexandrov (1)]: 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 (3), 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 (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 (4). Meeks showed (5) there are no examples with a single end. The unduloids themselves are the only examples with two ends (4). Kapouleas (6) has constructed examples (near the limit of zero necksize) 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 Fig. 1). In light of the trousers decomposition for surfaces, triunduloids can be seen as the building blocks for more complicated almost embedded CMC surfaces (7). Our main result determines explicitly the moduli space of triunduloids with labeled ends, up to Euclidean motions. Because 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.THEOREM. The proof of the theorem has three parts. First we define the classifying map from triunduloids to spherical triples, and observe that it is proper; then we prove it is injective; and finally we show it is surjective.To define the classifying map, we use the fact that any triunduloid has a reflection symmetry that decomposes the surface into mirror-image halves (8). Each half is simply connected, so Lawson's construction (9) gives a conjugate cousin minimal surface in the three-sphere. Using observations of Karcher (10), we find that its boundary projects under the Hopf map to the desired spherical triple. The composition of these steps defines our classifying map. ...

Geometry and topology of some overdetermined elliptic problems

“…The development of the technique of surface gluing and the construction of constant mean curvature surfaces with Delaunay ends has led to powerful methods in Geometric Analysis, see the paper of N. Kapouleas [22] and the papers of R. Mazzeo, F. Pacard and D. Pollack [26,27]. A related situation is the study of coplanar end surfaces, see the papers by C. Cosin and A. Ros [7], K. Große-Brauckmann, R. Kusner and J. Sullivan [20] and that of these authors joint with N. Korevaar and J. Ratzkin [19]. An other very powerful tool in Geometric Analysis is the use (often for comparison with the maximum principle) of the catenoid, a very well known unbounded minimal surface of revolution.…”

Constant mean curvature surfaces with Delaunay ends

“…and so the condition H = 1 leads to the equation (3)(4) pu-^l + pb + il + pff/^O, while the condition H = -1 yields (3.5) ^_I ( l + /9 2 ) _ (1 + p 2 ) 3/2 = 0 P There are two special solutions of (3.4) that can be determined immediately. The first is the constant solution pi = 1, the cylindrical graph of which is the cylinder of radius 1.…”

“…Recently, Kusner, Grosse-Brauckmann and Sullivan have shown that modulo Euclidean motions, .Mo,3 is a 3-ball [4]; it is plausible from their work that this particular moduli space contains no degenerate elements.…”

Foams and Bubbles: Geometry and Simulation

“…These force vectors 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 [5,6].…”