Abstract
We derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil, \(\widetilde{SL}_2\), and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. Using the spectral properties of the corresponding Dirac operators, we find analogs of the Willmore functional for surfaces in these geometries.
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Original Russian Text Copyright © 2005 Berdinsky D. A. and Taimanov I. A.
The authors were supported by the Russian Foundation for Basic Research (Grant 03-01-00403) and the RAS Program “Mathematical Methods in Nonlinear Dynamics.”
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 1248–1264, November–December, 2005.
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Berdinsky, D.A., Taimanov, I.A. Surfaces in Three-Dimensional Lie Groups. Sib Math J 46, 1005–1019 (2005). https://doi.org/10.1007/s11202-005-0096-9
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DOI: https://doi.org/10.1007/s11202-005-0096-9