Constant mean curvature (CMC) surfaces in space forms can be described by their associated C*-family of flat SL(2, C)-connections del(lambda). In this paper, we consider the asymptotic behavior (for lambda -> 0) of the gauge equivalence classes of del(lambda) for compact CMC surfaces of genus g >= 2. We prove (under the assumption of simple umbilics) that the asymptotic behavior of the traces of the monodromy representation of del(lambda) determines the conformal type as well as the Hopf differential locally in the Teichmuller space.
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