We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane $\boldsymbol{C}^2$ by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in $\boldsymbol{C}^2$.
Publié le : 2006-12-14
Classification:
Legendre curve,
Lagrangian immersion,
Hamiltonian-minimal,
elastica,
minimal immersion,
Lagrangian tori with constant mean curvature,
Lagrangian angle map,
53D12,
53C40,
53C42,
53B25
@article{1170347690,
author = {Castro, Ildefonso and Chen, Bang-Yen},
title = {Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves},
journal = {Tohoku Math. J. (2)},
volume = {58},
number = {1},
year = {2006},
pages = { 565-579},
language = {en},
url = {http://dml.mathdoc.fr/item/1170347690}
}
Castro, Ildefonso; Chen, Bang-Yen. Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves. Tohoku Math. J. (2), Tome 58 (2006) no. 1, pp. 565-579. http://gdmltest.u-ga.fr/item/1170347690/