The asymptotic behavior of the monodromy representations of the associated families of compact CMC surfaces

TL;DR

This paper studies the asymptotic behavior of monodromy representations associated with compact constant mean curvature surfaces, revealing how their traces determine conformal type and Hopf differential in the Teichmüller space.

Contribution

It establishes the link between the asymptotics of monodromy traces and the conformal and differential geometric data of CMC surfaces, under simple umbilics.

Findings

Asymptotic traces determine conformal type.

Asymptotic traces determine Hopf differential.

Results apply to genus g≥2 surfaces with simple umbilics.

Abstract

Constant mean curvature (CMC) surfaces in space forms can be described by their associated $\mathbb C^*$-family of flat $SL(2,\mathbb C)$-connections $\nabla^\lambda$. In this paper we consider the asymptotic behavior (for $\lambda\to0$) of the gauge equivalence classes of $\nabla^\lambda$ for compact CMC surfaces of genus $g\geq2.$ We prove (under the assumption of simple umbilics) that the asymptotic behavior of the traces of the monodromy representation of $\nabla^{\lambda}$ determines the conformal type as well as the Hopf differential locally in the Teichm\"uller space.