Toroidal Soap Bubbles: Constant Mean Curvature Tori in $$S^3$$ and $$R^3$$

Emma Carberry (Sydney)

Abstract

Constant mean curvature (CMC) tori in $$S^3$$, $$R^3$$ or $$H^3$$ are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes and whether tori can be deformed. I will address these questions for the spherical and Euclidean cases, using Whitham deformations.

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