On the Willmore problem for surfaces with symmetry

2021/03/17 by Kusner, Rob, Wang, Peng · 2 citations
#53A05 #53A10 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.09432

Abstract

The Willmore Problem seeks the surface in \mathbb S³⊂\mathbb R⁴ of a given topological type minimizing the squared-mean-curvature energy W = ∫ |H_ℝ⁴|² = area + ∫ H_\mathbbS³². The longstanding Willmore Conjecture that the Clifford torus minimizes W among genus-1 surfaces is now a theorem of Marques and Neves [19], but the general conjecture [10] that Lawson's [16] minimal surface ξ_g,1⊂\mathbb S³ minimizes W among surfaces of genus g>1 remains open. Here we prove this conjecture under the additional assumption that the competitor surfaces M⊂\mathbb S³ share the ambient symmetries of ξ_g,1. Specifcally, we show each Lawson surface ξ_m,k satisfies the analogous W-minimizing property under a somewhat smaller symmetry group G_m,k

Cited by

• Classification of Willmore 2-spheres in Sⁿ
• A new family of minimal surfaces of even genus in the three-dimensional sphere

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