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arXiv · 2506.20142

Application of Chern-Simons gauge theory to the enclosed volume of constant mean curvature surfaces in the 3-sphere

Lynn Heller·Sebastian Heller·Martin Traizet
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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 $\mathbb S^3$ 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 $g\geq2.$

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Lynn Heller, Sebastian Heller, Martin Traizet. 2025-06-25. Application of Chern-Simons gauge theory to the enclosed volume of constant mean curvature surfaces in the 3-sphere. https://arxiv.org/abs/2506.20142

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