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The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

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Abstract

Let F denote a surface with boundary ∂ F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve ωo parametrizing ∂F, we will get a manifold Ω of real dimension 6g−3, such that any ω∈Ω bounds some Fω and any local deformation\(\tilde F\) of F is conformally equivalent to just one Fω for ω∈Ω.

This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {Fω|ω∈Ω} cover all possible deformations of F at all.

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  1. Mathematisches Institut der RUB, Universitätsstr. 150, D-4630, Bochum 1

    Reinhold Böhme

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  1. Reinhold Böhme
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Böhme, R. The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces. Manuscripta Math 57, 205–223 (1987). https://doi.org/10.1007/BF02218081

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  • DOI: https://doi.org/10.1007/BF02218081

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