First explicit constrained Willmore minimizers of non-rectangular conformal class (Q2039553)

In this interesting paper the authors study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes \((0, b)\) with \(b \sim 1\) the homogeneous tori \(f^ b\) are known to be the unique constrained Willmore minimizers (up to invariance). The authors generalize this result and show that the candidates constructed in [\textit{L. Heller} and \textit{F. Pedit}, in: Willmore energy and Willmore conjecture. Boca Raton, FL: CRC Press. 119--138 (2018; Zbl 1384.53007)] are indeed constrained Willmore minimizers in certain non-rectangular conformal classes \((a, b)\). It is shown that the minimal Willmore energy \(\omega(a, b)\) is real analytic and concave in \(a \in (0, a^b)\) for some \(a^b > 0\) and fixed \(b \sim 1\), \(b \ne 1\).