The conformal theory of Alexandrov embedded constant mean curvature surfaces in $R^3$

TL;DR

This paper develops a gluing theorem for constructing new constant mean curvature surfaces in R^3 and explores the structure of their moduli space, revealing its complex topology and conformal properties.

Contribution

It introduces a gluing method for nondegenerate CMC surfaces and characterizes the conformal structure map, showing its surjectivity for genus zero surfaces.

Findings

New gluing theorem for CMC surfaces

Surjectivity of the conformal structure map for genus zero

Complex topological structure of the CMC moduli space

Abstract

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 of this paper, we develop the consequences of this result and (at least partially) characterize the image of the map which associates to each complete, Alexandrov-embedded CMC surface with finite topology its associated conformal structure, which is a compact Riemann surface with a finite number of punctures. In particular, we show that this `forgetful' map is surjective when the genus is zero. This proves in particular that the CMC moduli space has a complicated topological structure. These latter results are closely related to recent work of Kusner \cite{Ku}.