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Classification of a family of Hamiltonian-stationary Lagrangian submanifolds in C$^{n}$
Bang-Yen Chen
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 82, Number 9
(2006), 173-178.
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.
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Permanent link to this document: http://projecteuclid.org/euclid.pja/1165244967
Mathematical Reviews number (MathSciNet): MR2293505
Digital Object Identifier: doi:10.3792/pjaa.82.173
Zentralblatt MATH identifier: 1128.53052
References
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Mathematical Reviews (MathSciNet): MR2097394
Digital Object Identifier: doi:10.1307/mmj/1100623409
Project Euclid: euclid.mmj/1100623409
A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces, Tohoku Math. J. (2) 55 (2003), no. 4, 583--610.
Mathematical Reviews (MathSciNet): MR2017227
Digital Object Identifier: doi:10.2748/tmj/1113247132
Project Euclid: euclid.tmj/1113247132
Zentralblatt MATH: 1062.53053
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Mathematical Reviews (MathSciNet): MR1986315
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Zentralblatt MATH: 1042.53004
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Mathematical Reviews (MathSciNet): MR2245252
Digital Object Identifier: doi:10.1007/s10114-005-0690-6
I. Castro and B.-Y. Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. 58 (2006), 565--579.
Mathematical Reviews (MathSciNet): MR2297200
Digital Object Identifier: doi:10.2748/tmj/1170347690
Project Euclid: euclid.tmj/1170347690
Zentralblatt MATH: 05244429
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Mathematical Reviews (MathSciNet): MR1611051
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Zentralblatt MATH: 0896.53039
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Mathematical Reviews (MathSciNet): MR2247872
B.-Y. Chen, Geometry of submanifolds, Dekker, New York, 1973.
Mathematical Reviews (MathSciNet): MR353212
Zentralblatt MATH: 0262.53036
B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. (2) 49 (1997), no. 2, 277--297.
Mathematical Reviews (MathSciNet): MR1447186
Digital Object Identifier: doi:10.2748/tmj/1178225151
Project Euclid: euclid.tmj/1178225151
Zentralblatt MATH: 0877.53041
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Mathematical Reviews (MathSciNet): MR2222169
Digital Object Identifier: doi:10.2996/kmj/1143122389
Project Euclid: euclid.kmj/1143122389
Zentralblatt MATH: 1110.53061
B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257--266.
Mathematical Reviews (MathSciNet): MR346708
Digital Object Identifier: doi:10.2307/1996914
JSTOR: links.jstor.org
Zentralblatt MATH: 0286.53019
F. Hélein and P. Romon, Hamiltonian stationary Lagrangian surfaces in $\mathbf C\sp 2$, Comm. Anal. Geom. 10 (2002), no. 1, 79--126.
Mathematical Reviews (MathSciNet): MR1894142
F. Hélein and P. Romon, Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions, Comment. Math. Helv. 75 (2000), no. 4, 668--680.
Mathematical Reviews (MathSciNet): MR1789181
Digital Object Identifier: doi:10.1007/s000140050144
Zentralblatt MATH: 0973.53065
A. E. Mironov, On Hamiltonian-minimal and minimal Lagrangian submanifolds in $\mathbf C\sp n$ and $\mathbf C\rm P\sp n$, Dokl. Akad. Nauk 396 (2004), no. 2, 159--161.
Mathematical Reviews (MathSciNet): MR2115837
Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), no. 2, 501--519.
Mathematical Reviews (MathSciNet): MR1062973
Digital Object Identifier: doi:10.1007/BF01231513
R. Aiyama, Lagrangian surfaces with circle symmetry in the complex 2-space, Michigan Math. J. 52 (2004), 491--506.
Mathematical Reviews (MathSciNet): MR2097394
Digital Object Identifier: doi:10.1307/mmj/1100623409
Project Euclid: euclid.mmj/1100623409
A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces. Tohoku Math. J. 55 (2003), 583--610.
Mathematical Reviews (MathSciNet): MR2017227
Digital Object Identifier: doi:10.2748/tmj/1113247132
Project Euclid: euclid.tmj/1113247132
Zentralblatt MATH: 1062.53053
H. Anciaux, Construction of many \H surfaces in Euclidean four-space, Calc. of Var. 17 (2003), 105--120.
Mathematical Reviews (MathSciNet): MR1986315
Digital Object Identifier: doi:10.1007/s00526-002-0161-1
Zentralblatt MATH: 1042.53004
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.
Mathematical Reviews (MathSciNet): MR2245252
Digital Object Identifier: doi:10.1007/s10114-005-0690-6
I. Castro and B.-Y. Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. 58 (2006), 565--579.
Mathematical Reviews (MathSciNet): MR2297200
Digital Object Identifier: doi:10.2748/tmj/1170347690
Project Euclid: euclid.tmj/1170347690
Zentralblatt MATH: 05244429
I. Castro and F. Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in $\mathbf C\sp 2$, Compositio Math. 111 (1998), 1--14.
Mathematical Reviews (MathSciNet): MR1611051
Digital Object Identifier: doi:10.1023/A:1000332524827
Zentralblatt MATH: 0896.53039
I. Castro, H. Li and F. Urbano, \H submanifolds in complex space forms, to appear in Pacific J. Math.
Mathematical Reviews (MathSciNet): MR2247872
B.-Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973.
Mathematical Reviews (MathSciNet): MR353212
Zentralblatt MATH: 0262.53036
B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997), 277--297.
Mathematical Reviews (MathSciNet): MR1447186
Digital Object Identifier: doi:10.2748/tmj/1178225151
Project Euclid: euclid.tmj/1178225151
Zentralblatt MATH: 0877.53041
B.-Y. Chen, Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves, Kodai Math. J. 29 (2006), 84--112.
Mathematical Reviews (MathSciNet): MR2222169
Digital Object Identifier: doi:10.2996/kmj/1143122389
Project Euclid: euclid.kmj/1143122389
Zentralblatt MATH: 1110.53061
B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257--266.
Mathematical Reviews (MathSciNet): MR346708
Digital Object Identifier: doi:10.2307/1996914
JSTOR: links.jstor.org
Zentralblatt MATH: 0286.53019
F. Hélein and P. Romon, Hamiltonian stationary Lagrangian surfaces in $\mathbf C\sp 2$, Comm. Anal. Geom. 10 (2002), 79--126.
Mathematical Reviews (MathSciNet): MR1894142
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.
Mathematical Reviews (MathSciNet): MR1789181
Digital Object Identifier: doi:10.1007/s000140050144
Zentralblatt MATH: 0973.53065
A. E. Mironov, On Hamiltonian-minimal and minimal Lagrangian submanifolds in \cn and $CP^n$, Dokl. Akad. Nauk 396 (2004), 159--161.
Mathematical Reviews (MathSciNet): MR2115837
Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kaehler manifolds, Invent. Math. 101 (1990), 501--519.
Mathematical Reviews (MathSciNet): MR1062973
Digital Object Identifier: doi:10.1007/BF01231513
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