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

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; they are classified using their asymptotic necksizes. We work in a class slightly more general than embedded surfaces, namely immersed surfaces which bound an immersed three-manifold, as introduced by Alexandrov.Comment: LaTeX, 22 pages, 2 figures (8 ps files); full version of our announcement math.DG/9903101; final version (minor revisions) to appear in Crelle's J. reine angew. Mat

Coplanar constant mean curvature surfaces

We consider constant mean curvature surfaces of 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 in arXiv:math.DG/0102183. 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: we construct and classify the entire family of these genus-zero coplanar constant mean curvature surfaces.Comment: 35 pages, 10 figures; minor revisions including one new figure; to appear in Comm. Anal. Geo

Constant mean curvature surfaces with three ends

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.Comment: LaTex, 4 pages, 1 postscript figur

Coplanar k-unduloids are nondegenerate

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the coplanar classifying map is an analytic diffeomorphism.Comment: 19 pages, no figures; improvements to expositio

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to appear in Geometriae Dedicat

On the moduli spaces of embedded constant mean curvature surfaces with three or four ends

We are interested in explicitly parametrizing the moduli spaces Mg,k of embedded surfaces in R3 with finite genus g and a finite number of ends k having constant mean curvature. By rescaling we may assume this constant is 1, the mean curvature of the unit sphere. Two surfaces in R3 are indentified as points in Mg,k if there is isometry of R3 carrying one surface to the other. Moreover, we shall include in Mg,k a somewhat larger class of constant mean curvature (cmc) surfaces, the Alexandrov embedded surfaces, which are immersed surfaces bounding immersions of handlebodies into R3. The structure of these moduli spaces is known: they are finite dimensional real analytic varieties [KMP], but only a few of them are understood completely: the only embedded compact cmc surface is a round sphere [A], so Mg,0 is either a point (g = 0) or empty (g> 0); Mg,1 is empty, since there are no 1-ended examples [M]; and 2-ended examples are necessarily the Delaunay unduloids [KKS], which are simply-periodic surfaces of revolution whose minimal radius or necksize ρ ∈ (0, 1] parametrizes M0,2

MÖBIUS ENERGIES FOR KNOTS AND LINKS, SURFACES AND SUBMANIFOLDS

There has been recent interest in knot energies, especially those which are invariant under Möbius transformations of space. We describe computer experiments with such energies, and discuss ways of extending these to energies for higher-dimensional submanifolds. The appendix gives a table of computed energy-minimizing knots and links through eight crossings