# Isothermic constrained Willmore tori in 3-space

## TL;DR

This paper classifies all isothermic constrained Willmore tori in 3-space with Willmore energy below 8π, showing they are either homogeneous or 2-lobe Delaunay tori, and establishes non-degeneracy for certain classes.

## Contribution

It provides a complete classification of low-energy isothermic constrained Willmore tori in 3-space, identifying the only possible types and their properties.

## Key findings

- Homogeneous and 2-lobe Delaunay tori are the only such tori below 8π energy.
- Every constrained Willmore torus with energy below 8π and non-rectangular conformal class is non-degenerated.
- The classification narrows the understanding of constrained Willmore tori in 3-space.

## Abstract

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $8\pi$. In particular, every constrained Willmore torus with Willmore energy below $8\pi$ and non-rectangular conformal class is non-degenerated.