Conformal Willmore tori in ℝ4

Abstract

Abstract For every two-dimensional torus T 2 {T^{2}} and every k ∈ \in ℕ \mathbb{N} , k ≥ 3 {k\geq 3} , we construct a conformal Willmore immersion f : T 2 T^{2} → \to ℝ 4 \mathbb{R}^{4} with exactly one point of density k and Willmore energy 4πk. Moreover, we show that the energy value 8 ⁢ π {8\pi} cannot be attained by such an immersion. Additionally, we characterize the branched double covers T 2 T^{2} → \to S 2 S^{2} × \times { \{ 0 } \} as the only branched conformal immersions, up to Möbius transformations of ℝ 4 {\mathbb{R}^{4}} , from a torus into ℝ 4 {\mathbb{R}^{4}} with at least one branch point and Willmore energy 8 ⁢ π {8\pi} . 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 {8\pi} .