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

TL;DR

This paper derives a formula linking the enclosed volume of constant mean curvature surfaces in the 3-sphere to Chern-Simons gauge theory and the Willmore functional, demonstrating its effectiveness through examples.

Contribution

It introduces a novel formula connecting the enclosed volume of CMC surfaces to Chern-Simons gauge theory and the Willmore functional, expanding geometric analysis tools.

Findings

The formula effectively computes enclosed volume for various CMC surfaces.

Application to surfaces of genus g ≥ 2 shows practical utility.

Demonstrates gauge invariance of the enclosed volume calculation.

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 $g\geq2.$