Tohoku Mathematical Journal
2006
December
SECOND SERIES VOL. 58, NO. 4

Tohoku Math. J.
58 (2006), 565-579

Title LAGRANGIAN SURFACES IN COMPLEX EUCLIDEAN PLANE VIA SPHERICAL AND HYPERBOLIC CURVES

Author Ildefonso Castro and Bang-Yen Chen

(Received April 25, 2005, revised February 14, 2006)
Abstract. 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 $\boldsymbol{C}^2$.

2000 Mathematics Subject Classification. Primary 53D12; Secondary 53C40, 53C42, 53B25.
Key words and phrases. Legendre curve, Lagrangian immersion, Hamiltonian-minimal, elastica, minimal immersion, Lagrangian tori with constant mean curvature, Lagrangian angle map.

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