Tohoku Mathematical Journal

Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

Ildefonso Castro and Bang-Yen Chen

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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 $\boldsymbol{C}^2$.

Article information

Tohoku Math. J. (2), Volume 58, Number 4 (2006), 565-579.

First available in Project Euclid: 1 February 2007

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Zentralblatt MATH identifier

Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53B25: Local submanifolds [See also 53C40]

Legendre curve Lagrangian immersion Hamiltonian-minimal elastica minimal immersion Lagrangian tori with constant mean curvature Lagrangian angle map


Castro, Ildefonso; Chen, Bang-Yen. Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves. Tohoku Math. J. (2) 58 (2006), no. 4, 565--579. doi:10.2748/tmj/1170347690.

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