## Tohoku Mathematical Journal

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

#### 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

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

Digital Object Identifier
doi:10.2748/tmj/1170347690

Mathematical Reviews number (MathSciNet)
MR2297200

Zentralblatt MATH identifier
1193.53172

#### 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

#### References

• R. Aiyama, Lagrangian surfaces in the complex 2-space, Proceedings of the Fifth International Workshop on Differential Geometry (Taegu, 2000), 25--29, Kyungpook Natl. Univ., Taegu, 2001.
• I. Castro and F. Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in $\bfitC^2$, Compositio Math. 111 (1998), 1--14.
• I. Castro and F. Urbano, On a minimal Lagrangian submanifold of $\bfitC^n$ foliated by spheres, Michigan Math. J. 45 (1999), 71--82.
• I. Castro and F. Urbano, Willmore surfaces of $\bfitR^4$ and the Whitney sphere, Ann. Global Anal. Geom. 19 (2001), 153--175.
• B.-Y. Chen, Riemannian geometry of Lagrangian submanifolds, Taiwanese J. Math. 5 (2001), 681--723.
• B.-Y. Chen, Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math. 99 (1997), 69--108.
• B.-Y. Chen and J.-M. Morvan, Géométrie des surfaces lagrangiennes de $\bfitC^2$, J. Math. Pures Appl. (9) 66 (1987), 321--335.
• D. Joyce, Special Lagrangian $m$-folds in $\bfitC^m$ with symmetries, Duke Math. J. 115 (2002), 1--51.
• F. Helein and P. Romon, Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions, Comment. Math. Helv. 75 (2000), 668--680.
• D. Hoffmann and R. Osserman, The geometry of the generalized Gauss map, Mem. Amer. Math. Soc. 28 (1980).
• J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential. Geom. 20 (1984), 1--22.
• D. F. Lawden, Elliptic functions and applications, Appl. Math. Sci. 80, Springer-Verlag, New York, 1989.
• Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), 501--519.
• J. B. Seaborn, Hypergeometric functions and their applications, Texts Appl. Math. 8, Springer-Verlag, New York, 1991.