×

Classification of a family of Hamiltonian-stationary Lagrangian submanifolds in \(\mathbb C^{n}\). (English) Zbl 1128.53052

Summary: A Lagrangian submanifold in the complex Euclidean \(n\)-space \(\mathbb C^n\) 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 \(\mathbb C^n\) which are Lagrangian \(H\)-umbilical.

MSC:

53D12 Lagrangian submanifolds; Maslov index
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] R. Aiyama, Lagrangian surfaces with circle symmetry in the complex two-space, Michigan Math. J. 52 (2004), no. 3, 491-506. · Zbl 1073.53070 · doi:10.1307/mmj/1100623409
[2] 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. · Zbl 1062.53053 · doi:10.2748/tmj/1113247132
[3] H. Anciaux, Construction of many Hamiltonian stationary Lagrangian surfaces in Euclidean four-space, Calc. Var. Partial Differential Equations 17 (2003), no. 2, 105-120. · Zbl 1042.53004 · doi:10.1007/s00526-002-0161-1
[4] H. Anciaux, I. Castro and P. Romon, Lagrangian submanifolds foliated by \((n-1)\)-spheres in \(\mathbf R^ {2n}\), Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 4, 1197-1214. · Zbl 1105.53060 · doi:10.1007/s10114-005-0690-6
[5] I. Castro and B.-Y. Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. 58 (2006), 565-579. · Zbl 1193.53172 · doi:10.2748/tmj/1170347690
[6] I. Castro and F. Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in \(\mathbf C^ 2\), Compositio Math. 111 (1998), no. 1, 1-14. · Zbl 0896.53039 · doi:10.1023/A:1000332524827
[7] I. Castro, H. Li and F. Urbano, \H submanifolds in complex space forms, Pacific J. Math. (to appear). · Zbl 1129.53039 · doi:10.2140/pjm.2006.227.43
[8] B.-Y. Chen, Geometry of submanifolds , Dekker, New York, 1973. · Zbl 0262.53036
[9] B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. (2) 49 (1997), no. 2, 277-297. · Zbl 0877.53041 · doi:10.2748/tmj/1178225151
[10] B.-Y. Chen, Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves, Kodai Math. J. 29 (2006), no. 1, 84-112. · Zbl 1110.53061 · doi:10.2996/kmj/1143122389
[11] B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257-266. · Zbl 0286.53019 · doi:10.2307/1996914
[12] F. Hélein and P. Romon, Hamiltonian stationary Lagrangian surfaces in \(\mathbf C^ 2\), Comm. Anal. Geom. 10 (2002), no. 1, 79-126. · Zbl 1007.53060
[13] 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. · Zbl 0973.53065 · doi:10.1007/s000140050144
[14] A. E. Mironov, On Hamiltonian-minimal and minimal Lagrangian submanifolds in \(\mathbf C^ n\) and \(\mathbf C{\mathrm P}^ n\), Dokl. Akad. Nauk 396 (2004), no. 2, 159-161.
[15] Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), no. 2, 501-519. \beginthebibliography99 · Zbl 0721.53060 · doi:10.1007/BF01231513
[16] R. Aiyama, Lagrangian surfaces with circle symmetry in the complex 2-space, Michigan Math. J. 52 (2004), 491-506. · Zbl 1073.53070 · doi:10.1307/mmj/1100623409
[17] A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces. Tohoku Math. J. 55 (2003), 583-610. · Zbl 1062.53053 · doi:10.2748/tmj/1113247132
[18] H. Anciaux, Construction of many \H surfaces in Euclidean four-space, Calc. of Var. 17 (2003), 105-120. · Zbl 1042.53004 · doi:10.1007/s00526-002-0161-1
[19] 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. · Zbl 1105.53060 · doi:10.1007/s10114-005-0690-6
[20] I. Castro and B.-Y. Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. 58 (2006), 565-579. · Zbl 1193.53172 · doi:10.2748/tmj/1170347690
[21] I. Castro and F. Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in \(\mathbf C^ 2\), Compositio Math. 111 (1998), 1-14. · Zbl 0896.53039 · doi:10.1023/A:1000332524827
[22] I. Castro, H. Li and F. Urbano, \H submanifolds in complex space forms , to appear in Pacific J. Math. · Zbl 1129.53039 · doi:10.2140/pjm.2006.227.43
[23] B.-Y. Chen, Geometry of Submanifolds , Dekker, New York, 1973. · Zbl 0262.53036
[24] B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997), 277-297. · Zbl 0877.53041 · doi:10.2748/tmj/1178225151
[25] B.-Y. Chen, Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves, Kodai Math. J. 29 (2006), 84-112. · Zbl 1110.53061 · doi:10.2996/kmj/1143122389
[26] B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257-266. · Zbl 0286.53019 · doi:10.2307/1996914
[27] F. Hélein and P. Romon, Hamiltonian stationary Lagrangian surfaces in \(\mathbf C^ 2\), Comm. Anal. Geom. 10 (2002), 79-126. · Zbl 1007.53060
[28] 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. · Zbl 0973.53065 · doi:10.1007/s000140050144
[29] A. E. Mironov, On Hamiltonian-minimal and minimal Lagrangian submanifolds in \cn and \(CP^n\), Dokl. Akad. Nauk 396 (2004), 159-161.
[30] Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kaehler manifolds, Invent. Math. 101 (1990), 501-519. · Zbl 0721.53060 · doi:10.1007/BF01231513
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.