Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions |
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Authors: | F Hélein P Romon |
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Institution: | (1) Institut Universitaire de France, CMLA, ENS de Cachan, 61 avenue du Président Wilson, F-94235 Cachan Cedex, France, e-mail: helein@cmla.ens-cachan.fr, FR;(2) Institut Universitaire de France, CMLA, ENS de Cachan, 61 avenue du Président Wilson, F-94235 Cachan Cedex, France, and Université de Marne-la-Vallée, 5, boulevard Descartes, Champs sur Marne, F-77454 Marne-la-Vallée Cedex 2, France, e-mail: romon@cmla.ens-cachan.fr, FR |
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Abstract: | We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies
an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using
quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach
to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein
bottles and show they self-intersect.
Received: January 25, 2000. |
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Keywords: | , Lagrangian surfaces, Weierstrass representation, Dirac equation, minimal surfaces, variational problem with constraint, |
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