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 $a{\sim }_b{0}^{+}$ deforming the homogenous 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 $b\sim 1$, $b\ne 1$ and $a{\sim }_b{0}^{+}$ 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 $f^b$. Indeed we show in \cite{HelNdi1} that these candidates minimize the Willmore energy in their respective conformal class for $b\sim 1$, $b\ne 1$ and $a{\sim }_b{0}^{+}.$

arXiv comment35 pages, originally a section of arXiv:1710.00533, we decided to split the paper due to length