Candidates for non-rectangular constrained Willmore minimizers. (English) Zbl 1470.53013
Summary: For every \(b>1\) fixed, we explicitly construct 1-dimensional families of embedded constrained Willmore tori parametrized by their conformal class \((a,b)\) with \(a\sim_b0^+\) deforming the homogeneous torus \(f^b\) of conformal class \((0,b)\). The variational vector field at \(f^b\) is hereby given by a non-trivial zero direction of a penalized Willmore stability operator which we show to coincide with a double point of the corresponding spectral curve. Further, we characterize for \(b\sim 1\), \(b\neq 1\) and \(a\sim_b0^+\) the family obtained by opening the “smallest” double point on the spectral curve which is heuristically the direction with the smallest increase of Willmore energy at \(f^b\). Indeed we show in [Adv. Math. 386, Article ID 107804, 47 p. (2021; Zbl 1483.53012)] that these candidates minimize the Willmore energy in their respective conformal class for \(b\sim 1\), \(b\neq 1\) and \(a\sim_b0^+\).
MSC:
| 53A31 | Differential geometry of submanifolds of Möbius space |
| 49Q10 | Optimization of shapes other than minimal surfaces |
Keywords:
Willmore energy; Willmore functional; penalized Willmore stability operator; spectral curveCitations:
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