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

Constant mean curvature surfaces in S3S3 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 Hitchin (1990), Pinkall and Sterling (1989) and for genus 2 surfaces (Heller, 2014) to CMC surfaces in S3S3 of genus g=k⋅lg=k⋅l with commuting Zk+1Zk+1 and Zl+1Zl+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 Heller (2015) and prove the short time existence of this flow for certain families of initial surfaces. In this way we obtain countably many 1−1−parameter families of new CMC surfaces of higher genus with prescribed branch points and prescribed umbilics.