Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero

Abstract

We construct the entire three-parameter family of embedded constant mean curvature surfaces with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the necksizes of the three ends. Because our surfaces are transcendental, and are not described by any ordinary di¤erential equation, it is remarkable to obtain such an explicit determination of their moduli space.

Isometric immersions into 3-dimensional homogeneous manifolds

“…In particular we get a local isometric correspondence between minimal surfaces in the Heisenberg group Nil 3 (with its standard metric) and CMC 1 2 surfaces in H 2 × R. In this case we have θ = π 2 , which makes this correspondence similar to the conjugate cousin correspondence in space forms (see [GBKS03], [Kar05]). We compute some examples: the sister surface of the rotational minimal surface of equation z = 0 in Nil 3 is a graph over H 2 in H 2 × R invariant by a vertical rotation; the sister surface of the translational minimal surface of equation z = xy 2 in Nil 3 is a graph over H 2 in H 2 × R invariant by a hyperbolic translation.…”

“…This correspondence has nice geometric properties, and is useful to construct CMC 1 surfaces in H 3 with some prescribed geometric properties starting from a solution of a Plateau problem in R 3 (see for example [Kar05], [Dan06]). In the same way, the conjugate cousin correspondence between minimal surfaces in S 3 and CMC 1 surfaces in R 3 was used to construct CMC 1 triunduloids in R 3 ( [GBKS03]). …”

Construction of compact constant mean curvature hypersurfaces with topology

“…In the case of complete noncompact constant mean curvature surfaces, the moduli space of such surfaces is now fairly well understood (in the genus 0 case). Then, many examples of such surfaces are produced in [9] and [16] and a classification of embedded constant mean curvature surfaces with three ends is given in [3]. However, the set of compact constant mean curvature is not so well understood.…”

Coplanar k-Unduloids Are Nondegenerate

“…Here Σ + := f −1 (M + ) and the immersions f and f are related by the first-order cousin equation GKS1]. From this equation, we see that f is also an isometry, and that the normal ν of M + left translates to the normal ν = f ν of M + .…”

“…No classification results are yet available for CMC surfaces with higher genus or noncoplanar ends, but we do know [GKS1,GKS2] that M ′ k is homeomorphic to a certain connected (2k −3)-manifold D k of spherical metrics on the disk.…”