Rectangular constrained Willmore minimizers and the Willmore conjecture

TL;DR

This paper proves that a specific family of Delaunay tori uniquely minimizes the Willmore energy in their conformal class, providing an alternative proof of the Willmore conjecture and suggesting a generalization approach.

Contribution

It introduces a new proof of the Willmore conjecture using Delaunay tori and proposes a method extendable to higher codimensions under certain classifications.

Findings

Delaunay tori uniquely minimize Willmore energy in their conformal class.

Provides an alternative proof of the Willmore conjecture in 3-space.

Suggests a generalization framework for higher codimensions.

Abstract

We show that the well-known family of $2$-lobed Delaunay tori $\;f^b\;$ in $\;S^3,\;$ parametrized by $\;b \in \mathbb R_{\geq1},\;$ uniquely minimizes the Willmore energy among all immersions from tori into $3$-space of conformal class $\;(a, b)\;$. As a corollary we obtain an alternate proof of the Willmore conjecture in $3$-space. This new strategy can be generalized to arbitrary codimensions provided a classification of isothermic constrained Willmore tori is possible and all $\;f^b\;$ remain stable in all codimensions.