First explicit constrained Willmore minimizers of non-rectangular conformal class

We study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes (0,b) with b∼1 the homogeneous tori fb are known to be the unique constrained Willmore minimizers (up to invariance). In this paper we generalize this result and show that the candidates constructed in [14] are indeed constrained Willmore minimizers in certain non-rectangular conformal classes (a,b). Difficulties arise from the fact that these minimizers are non-degenerate for a≠0 but smoothly converge to the degenerate homogeneous tori fb as a⟶0. As a byproduct of our arguments, we show that the minimal Willmore energy ω(a,b) is real analytic and concave in a∈(0,ab) for some ab>0 and fixed b∼1, b≠1.