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Spectral methods for higher genus constant mean curvature surfaces
This thesis investigates two possible versions of a "spectral curve" construction for compact constant mean curvature (CMC) surfaces of genus g > 1 in [special characters omitted]. The first version uses the holonomy spectral curve which was originally formulated for tori in [special characters omitted]. In order to make sense of the definition of this curve for a higher genus surface M, we must assume that the holonomy is abelian, and in this case it is shown that M must be a branched immersion factoring holomorphically through a CMC torus which can be located naturally in the Jacobian of M. The second version uses a curve defined as a double cover of M branched at the zeroes of the Hopf differential Q which coincides with that used originally by Hitchin to analyze the moduli space of stable bundles over M. We propose a method of defining a CMC immersion of this curve which has abelian holonomy and therefore, by the earlier result, factors through a naturally defined CMC torus. Along with the non-abelian holonomy of a certain meromorphic connection around the zeroes of Q, this data might provide effective moduli for M.
Gerding, Aaron, "Spectral methods for higher genus constant mean curvature surfaces" (2011). Doctoral Dissertations Available from Proquest. AAI3482626.