# Candidates for non-rectangular constrained Willmore minimizers

## TL;DR

This paper constructs explicit families of constrained Willmore tori near homogeneous tori, analyzing their stability and spectral curve properties to identify potential minimizers of the Willmore energy within their conformal classes.

## Contribution

It provides explicit constructions of constrained Willmore tori parametrized by conformal class and characterizes their stability and spectral properties, advancing understanding of Willmore minimizers.

## Key findings

- Constructed families of constrained Willmore tori near homogeneous tori.
- Identified stability directions via spectral curve analysis.
- Characterized energy-minimizing candidates in specific conformal classes.

## Abstract

For every $\;b>1\;$ fixed, we explicitly construct $1$-dimensional families of embedded constrained Willmore tori parametrized by their conformal class $\;(a,b)$\; with $\; a \sim_b 0^+\;$ deforming the homogenous torus \;$f^b$ of conformal class \;$(0,b).$ The variational vector field at $f^b$ is hereby given by a non-trivial zero direction of a penalized Willmore stability operator which we show to coincide with a double point of the corresponding spectral curve. Further, we characterize for $b \sim 1$, $b \neq 1$ and $a \sim_b 0^+$ the family obtained by opening the "smallest" double point on the spectral curve which is heuristically the direction with the smallest increase of Willmore energy at $f^b$. Indeed we show in \cite{HelNdi1} that these candidates minimize the Willmore energy in their respective conformal class for $b \sim 1$, $b \neq 1$ and $a \sim_b 0^+.$