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 ${\mathbb{C}}^{\ast }$-family of flat $SL(2,\mathbb{C})$-connections ${\nabla }^{\lambda }$. In this paper we consider the asymptotic behavior (for $\lambda \to 0$) of the gauge equivalence classes of ${\nabla }^{\lambda }$ for compact CMC surfaces of genus $g\ge 2.$ 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 Teichmüller space.

6 pages

References in corpus (1)

• The spectral curve theory for $(k,l)-$symmetric CMC surfaces