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

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.