Rectangular constrained Willmore minimizers and the Willmore conjecture

Abstract

We show that the well-known family of $2$-lobed Delaunay tori $f^b$ in $S^3,$ parametrized by $b\in {\mathbb{R}}_{\ge 1},$ 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.

Gap in the proof in section 2. Sections 3 and 4 are not impacted and will appear as separate preprints

Cited by in corpus (1)

• Higher Regularity of Weak Limits of Willmore Immersions I