Conformal Willmore tori in ℝ4

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