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.
| Identifer | oai:union.ndltd.org:UMASS/oai:scholarworks.umass.edu:dissertations-6333 |
| Date | 01 January 2011 |
| Creators | Gerding, Aaron |
| Publisher | ScholarWorks@UMass Amherst |
| Source Sets | University of Massachusetts, Amherst |
| Language | English |
| Detected Language | English |
| Type | text |
| Source | Doctoral Dissertations Available from Proquest |
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