Conformal Willmore Tori in $\mathbb{R}^4$

For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4\pi k$. 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 M\"obius 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 $8\pi$.