Conformal Willmore Tori in ℝ⁴

2015/06/30 by Lamm, Tobias, Schätzle, Reiner M.
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.09154

Abstract

For every two-dimensional torus T² and every k∈ ℕ, k≥ 3, we construct a conformal Willmore immersion f:T²→ ℝ⁴ with exactly one point of density k and Willmore energy 4πk. Moreover, we show that the energy value 8π cannot be attained by such an immersion. Additionally, we characterize the branched double covers T²→ S² × \0\ as the only branched conformal immersions, up to Möbius transformations of ℝ⁴, from a torus into ℝ⁴ with at least one branch point and Willmore energy 8π. Using a perturbation argument in order to regularize a branched double cover, we finally show that the infimum of the Willmore energy in every conformal class of tori is less than or equal to 8π.

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