First explicit constrained Willmore minimizers of non-rectangular conformal class

TL;DR

This paper extends the understanding of constrained Willmore minimizers for immersed tori in 3-space, demonstrating new minimizers in non-rectangular conformal classes and analyzing their energy properties.

Contribution

It generalizes known results by identifying new constrained Willmore minimizers in non-rectangular classes and studies their energy analyticity and concavity.

Findings

Constructed constrained Willmore minimizers in non-rectangular classes.

Proved the minimal Willmore energy is real analytic and concave in certain parameters.

Showed convergence of minimizers to degenerate tori as parameters approach zero.

Abstract

We study immersed tori in $3$-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes $\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to be the unique constrained Willmore minimizers (up to invariance). In this paper we generalize this result and show that the candidates constructed in \cite{HelNdi2} are indeed constrained Willmore minimizers in certain non-rectangular conformal classes $\;(a,b).\;$ Difficulties arise from the fact that these minimizers are non-degenerate for $\;a \neq 0\;$ but smoothly converge to the degenerate homogenous tori $\;f^b\;$ as $\;a \longrightarrow 0.\;$ As a byproduct of our arguments, we show that the minimal Willmore energy $\;\omega(a,b)\;$ is real analytic and concave in $\;a \in (0, a^b)\;$ for some $\;a^b>0\;$ and fixed $\;b \sim 1,\;$ $b \neq 1.$