Constrained Willmore and CMC tori in the 3-sphere

TL;DR

This paper investigates the relationship between constrained Willmore tori and constant mean curvature (CMC) tori in the 3-sphere, showing that spectral curve structures can identify CMC tori among constrained Willmore tori.

Contribution

It demonstrates that a constrained Willmore torus in the 3-sphere is a CMC torus if its spectral curve has a CMC spectral curve structure, under mild conditions.

Findings

Spectral curves of CMC tori are hyperelliptic.

Constrained Willmore tori with CMC spectral curves are CMC in space forms.

Provides criteria to identify CMC tori via spectral data.

Abstract

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface \Sigma, such that f can be reconstructed in terms of algebraic data on \Sigma. Particularly interesting examples of constrained Willmore tori are the tori with constant mean curvature (CMC) in a 3-dimensional space form. It is shown in [14] and in [16] that the spectral curves of these tori are hyperelliptic. In this paper we show under mild conditions that a constrained Willmore torus f in the 3-sphere is a CMC torus in a 3-dimensional space form if its spectral curve has the structure of a CMC spectral curve.