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

TL;DR

This paper constructs conformal Willmore tori in four-dimensional space with specific energy and density properties, characterizes branched immersions, and establishes bounds on the Willmore energy for tori.

Contribution

It introduces explicit constructions of conformal Willmore tori with prescribed densities and energies, and characterizes branched conformal immersions with minimal energy.

Findings

Constructed Willmore tori with energy 4πk and a single density point.

Proved that energy 8π cannot be achieved by such immersions.

Showed the infimum of Willmore energy in any conformal class of tori is at most 8π.

Abstract

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 $8\pi$.