Tohoku Mathematical Journal

Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

Ildefonso Castro and Bang-Yen Chen

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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 $\boldsymbol{C}^2$.

Article information

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

Dates
First available in Project Euclid: 1 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1170347690

Digital Object Identifier
doi:10.2748/tmj/1170347690

Mathematical Reviews number (MathSciNet)
MR2297200

Zentralblatt MATH identifier
1193.53172

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.tmj/1170347690


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