A note on Willmore minimizing Klein bottles in Euclidean space

2016/06/15 by J. Hirsch, Jonas Hirsch, Hirsch, J. +3
Mathematics · #53A07 #53C42 #58J50 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.AP #math.DG #msc:53A07 #msc:53C42 #msc:58J50

paper · pdf · doi:10.48550/arxiv.1606.04745

8 pages

openalex publication_date 2016/06/15 · arxiv created 2017/02/28 · arxiv updated 2017/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that Lawson's bipolar surface τ_3,1 is after stereographic projection the unique minimizer among immersed Klein bottles in its conformal class. We conjecture that it actually is the unique minimizer among immersed Klein bottles into ℝⁿ, n≥ 4, whose existence the authors and P. Breuning proved in a previous paper.

Citations

• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

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• Conformal Geometric Inequalities on the Klein Bottle
• On the Willmore problem for surfaces with symmetry
• The Willmore problem for surfaces with symmetry
• Minimizers of the Willmore functional under fixed conformal class