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

2025/06/25 by Lynn Heller, Heller, Lynn, Sebastian Heller +3
Mathematics · Computer Science · #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2506.20142

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 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.

Citations

• Minimal surfaces and alternating multiple zetas
• Minimal reflection surfaces in \mathbb S³. Combinatorics of curvature lines and minimal surfaces based on fundamental pentagons
• Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere
• Fuchsian DPW potentials for Lawson surfaces
• Area Estimates for High genus Lawson surfaces via DPW
• Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line
• The Wess-Zumino term for a harmonic map

Related

• A Simons' type formula for cmc surfaces in homogeneous 3-manifolds
• Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton)
• Exploring the space of compact symmetric CMC surfaces
• Area bounds for constant mean curvature surfaces in hyperbolic 3-manifolds
• Simons type formulas for surfaces in Sol₃ and applications