arXiv research

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over Riemann surfac…

Researchers create annular translators for mean curvature flow.

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.

The paper proves compactness of metrics with isolated singularities on a sphere.

Compactness of metrics with positive sixth order Q-curvature on a sphere with punctures.

We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials p…

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; the…

We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-γ}}$ for $|X|$ large, when $A<0$ and $γ\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, fold…

All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family D_tau of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to parameter value tau element in R^-, are immersed and are called nodoids. The…

We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method of Cauchy data matching from \cite{MP}, cf. also \cite{MPP}. In the second part…