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

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in \cite{Hi, PiSt} and for genus $2$ surfaces \cite{He3} to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting $\mathbb Z_{k+1}$ and $\mathbb Z_{l+1}$ symmetries. We determine their associated family of flat connections via certain flat line bundle connections parametrized by the spectral curve. We generalize the flow on spectral data introduced in \cite{HeHeSch} and prove the short time existence of this flow for certain families of initial surfaces. In this way we obtain various families of new CMC surfaces of higher genus with prescribed branch points and prescribed umbilics.