TY - JOUR

T1 - Candidates for non-rectangular constrained Willmore minimizers

AU - Heller, Lynn

AU - Ndiaye, Cheikh Birahim

N1 - Funding Information: The first author wants to thank the Ministry of Science, Research and Art Baden-Wüttemberg, Germany and the European social Fund for supporting her research within the Margerete von Wrangell Programm. Further, both authors are indebted to the Baden-Württemberg foundation, Germany for supporting the project within the Eliteprogramm for Postdocs. Moreover, the second author thank the Deutsche Forschungsgemeinschaft (DFG), Germany for financial support through the project ”Fourth-order uniformization type theorem for 4-dimensional Riemannian manifolds” and the Schweizerischer Nationalfonds (SNF) for financial support through the project PP00P2_144669 .

PY - 2021/7

Y1 - 2021/7

N2 - 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 \(b \sim 1\), \(b \neq 1\) 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 \(b \sim 1\), \(b \neq 1\) and \(a \sim_b 0^+.\)

AB - 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 \(b \sim 1\), \(b \neq 1\) 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 \(b \sim 1\), \(b \neq 1\) and \(a \sim_b 0^+.\)

KW - math.DG

KW - 53A05, 53A30, 53C43

UR - http://www.scopus.com/inward/record.url?scp=85103419056&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1902.09572

DO - 10.48550/arXiv.1902.09572

M3 - Article

VL - 165

JO - J. Geom. Phys.

JF - J. Geom. Phys.

M1 - 104221

ER -