Equivariant constrained Willmore tori in the 3-sphere

TL;DR

This paper classifies equivariant constrained Willmore tori in the 3-sphere by analyzing their spectral curves, revealing a bound on spectral genus and providing a framework for understanding their geometric properties.

Contribution

It introduces a classification of equivariant constrained Willmore tori based on spectral curve genus, with a detailed analysis of their Möbius symmetries and spectral data.

Findings

Spectral curve of equivariant tori is a double cover of the complex plane.

Spectral genus of these tori is at most 3.

Classification based on spectral genus g.

Abstract

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of M\"obius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb C$ and classify equivariant constrained Willmore tori by the genus g of their spectral curve. In this case the spectral genus satisfies $g \leq 3.$