## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Classification of a family of Hamiltonian-stationary Lagrangian submanifolds in C$^{n}$

Bang-Yen Chen

#### Abstract

A Lagrangian submanifold in the complex Euclidean $n$-space ${\bf C}^n$ is called Hamiltonian-stationary if it is a critical point of the area functional restricted to (compactly supported) Hamiltonian variations. In this article, we classify the family of Hamiltonian-stationary Lagrangian submanifolds of ${\bf C}^n$ which are Lagrangian $H$-umbilical.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 9 (2006), 173-178.

Dates
First available in Project Euclid: 4 December 2006

https://projecteuclid.org/euclid.pja/1165244967

Digital Object Identifier
doi:10.3792/pjaa.82.173

Mathematical Reviews number (MathSciNet)
MR2293505

Zentralblatt MATH identifier
1128.53052

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index

#### Citation

Chen, Bang-Yen. Classification of a family of Hamiltonian-stationary Lagrangian submanifolds in C$^{n}$. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 9, 173--178. doi:10.3792/pjaa.82.173. https://projecteuclid.org/euclid.pja/1165244967

#### References

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• R. Aiyama, Lagrangian surfaces with circle symmetry in the complex 2-space, Michigan Math. J. 52 (2004), 491–506.
• A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces. Tohoku Math. J. 55 (2003), 583–610.
• H. Anciaux, Construction of many \H surfaces in Euclidean four-space, Calc. of Var. 17 (2003), 105–120.
• H. Anciaux, I. Castro and P. Romon, Lagrangian submanifolds foliated by $(n-1)$-spheres in $\mathbf E^{2n}$, Acta Math. Sinica, 22 (2006), 1197–1214.
• I. Castro and B.-Y. Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. 58 (2006), 565–579.
• I. Castro and F. Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in $\mathbf C\sp 2$, Compositio Math. 111 (1998), 1–14.
• I. Castro, H. Li and F. Urbano, \H submanifolds in complex space forms, to appear in Pacific J. Math.
• B.-Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973.
• B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997), 277–297.
• B.-Y. Chen, Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves, Kodai Math. J. 29 (2006), 84–112.
• B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257–266.
• F. Hélein and P. Romon, Hamiltonian stationary Lagrangian surfaces in $\mathbf C\sp 2$, Comm. Anal. Geom. 10 (2002), 79–126.
• F. Hélein and P. Romon, Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions, Comment. Math. Helv. 75 (2000), 668–680.
• A. E. Mironov, On Hamiltonian-minimal and minimal Lagrangian submanifolds in \cn and $CP^n$, Dokl. Akad. Nauk 396 (2004), 159–161.
• Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kaehler manifolds, Invent. Math. 101 (1990), 501–519.