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Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

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Abstract

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.

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Original Russian Text Copyright © 2007 Berdinsky D. A. and Taĭmanov I. A.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 3, pp. 496–511, May–June, 2007.

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Berdinsky, D.A., Taimanov, I.A. Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional. Sib Math J 48, 395–407 (2007). https://doi.org/10.1007/s11202-007-0043-z

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  • DOI: https://doi.org/10.1007/s11202-007-0043-z

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