Candidates for non-rectangular constrained Willmore minimizers
Abstract
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 b 0+\; deforming the homogenous torus \;fb of conformal class \;(0,b). The variational vector field at fb 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 1, b ≠ 1 and a b 0+ 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 fb. Indeed we show in HelNdi1 that these candidates minimize the Willmore energy in their respective conformal class for b 1, b ≠ 1 and a b 0+.
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