Application of Chern-Simons gauge theory to the enclosed volume of constant mean curvature surfaces in the 3-sphere
Abstract
Building on Hitchin's work of the Wess-Zumino-Witten term for harmonic maps into Lie groups, we derive a formula for the enclosed volume of a compact CMC surface f in S3 in terms of a holonomy on the Chern-Simons bundle and the Willmore functional. By construction the enclosed volume only depends on the gauge classes of the associated family of flat connections of f. In this paper we show in various examples the effectiveness of this formula, in particular for surfaces of genus g≥2.
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