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Constant mean curvature (CMC) surfaces in space forms can be described by their associated    C  ∗   ‐family of flat    SL  (  2  ,  C  )   ‐connections    ∇  λ   . In this paper, we consider the asymptotic behavior (for    λ  →  0   ) of the gauge equivalence classes of    ∇  λ    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    ∇  λ    determines the conformal type as well as the Hopf differential locally in the Teichmüller space.

The asymptotic behavior of the monodromy representation of the associated family of a compact CMC surface