Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

Details Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

2006-03-15

arxiv.org/abs/math/0603368v1

Mathematics

Differential Geometry

16 pages To be published in Tohoku Mathematical Journal

Scientific paper

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane 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 complex Euclidean plane.

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