  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.
