Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces

TL;DR

This paper studies the relationship between conformal structures and neck sizes of embedded constant mean curvature surfaces, showing the classifying map is proper and analyzing the implications for genus zero surfaces.

Contribution

It introduces a real analytic classifying map for CMC surfaces linking conformal structures and neck sizes, and analyzes its properties including properness and degree.

Findings

The classifying map is proper and real analytic.

For genus zero, all punctured spheres can be realized as CMC surfaces.

The degree of the classifying map is zero, implying an even number of surfaces per conformal type.

Abstract

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 surfaces of genus g with k (marked) punctures, the real analytic structure coming from the 3g-3+k local complex analytic coordinates on the Riemann moduli space r_{g,k}. Then the parabolic classifying map, Phi: M --> P, which assigns to a CMC surface its induced conformal structure and asymptotic necksizes, is a proper, real analytic map. It follows that Phi is closed and in particular has closed image. For genus g=0, this can be used to show that every conformal type of multiply punctured Riemann sphere occurs as a CMC surface, and -- under a nondegeneracy hypothesis -- that Phi has a well defined (mod 2) degree. This degree vanishes, so generically an even number of CMC surfaces realize any given conformal structure and asymptotic necksizes.