Rectangular constrained Willmore minimizers and the Willmore conjecture

2019/01/17 by Heller, Lynn, Heller, Sebastian, Ndiaye, Cheikh Birahim
#53A05 #53A30 #53C43 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.05664

Abstract

We show that the well-known family of 2-lobed Delaunay tori f^b in S³, parametrized by b ∈ \mathbb R_≥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.

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