Conformal Willmore tori in ℝ4

Tobias Lamm; Reiner M. Schätzle

doi:10.1515/crelle-2015-0101

Article

Conformal Willmore tori in ℝ⁴

Tobias Lamm and Reiner M. Schätzle

Published/Copyright: February 16, 2016

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 742

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 8⁢π 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 8⁢π. 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 8⁢π.

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Received: 2015-07-10

Revised: 2015-10-23

Published Online: 2016-02-16

Published in Print: 2018-09-01

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