Riemannian geometry of Lagrangian submanifolds

Taiwanese Journal of Mathematics

Volumn 5, Issue 4, 2001, Pages 681-723

  Chen, Bang Yen ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Complex space form; Finite type theory; H umbilical Lagrangian submanifold; Ideal immersion; Index; Lagrangian catenoid; Lagrangian submanifolds; Maslov class; Obstruction; Parallel submani; Pinching theorem; Stability; Totally real submanifold; Vanishing theorem

Indexed keywords

EID: 0037853659     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: 10.11650/twjm/1500574989     Document Type: Article

References (116)

1. D. E. Blair, F. Dillen, L. Verstraelen and L. Vrancken, Calabi curves as holomorphic Legendre curves and Chen's inequality, Kyungpook Math. J. 35 (1996), 407-416.
2. n, Math. Ann. 279 (1988), 599-620.
3. 3 satisfying Chen's equality, preprint.
4. V. Borrelli, B. Y. Chen and J. M. Morvan, Une caractérisation gómétrique de la sphère de Whitney, C. R. Acad. Sci. Paris Sér. J Math. 321 (1995), 1485-1490.
5. R. L. Bryant, Minimal Lagrangian submanifolds of Kaehler-Einstein manifolds, in: Differential Geometry and Differential Equations (Shanghai, 1985), Lecture Notes in Math. 1255, 1987, Springer, New York, pp. 1-12.
6. R. L. Bryant, Some examples of special Lagrangian tori, Adv. Theor. Math. Phys. 3 (1999), 83-90.
7. E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958), 407-438.
8. I. Castro, C. R. Montealegre and F. Urbano, Closed conformal vector fields and Lagrangian submanifolds in complex space forms, Pacific J. Math., to appear.
9. I. Castro, Lagrangian spheres in the complex Euclidean space satisfying a geometric equality, Geom. Dedicata 70 (1998), 197-208.
10. I. Castro, The Lagrangian version of a theorem of J. B. Meusnier, in: Proc. Summer School on Diff. Geom., Coimbra 1999, pp. 83-89.
11. I. Castro and F. Urbano, Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form, Tôhoku Math. J. 45 (1993), 656-582.
12. I. Castro and F. Urbano, New examples of minimal Lagrangian tori in the complex projective plane, Manuscripta Math. 85 (1994), 265-281.
13. I. Castro and F. Urbano, Twistor holomorphic Lagrangian surface in the complex projective and hyperbolic planes, Ann. Global Anal. Geom. 13 (1995), 59-67.
14. 2 Compositis Math. 111 (1998), 1-14.
15. n foliated by spheres, Michigan Math. J. 46 (1999), 71-82.
16. 2 J. Geom. Anal. 10 (2000), 243-255.
17. B. Y. Chen, Geometry of Submanifolds, Mercel Dekker, New York, 1973.
18. B. Y. Chen, Geometry of Submanifolds and its Applications, Science Univ. Tokyo, Tokyo, 1981.
19. B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publisher, New Jersey, 1984.
20. B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel), 60 (1993), 568-578.
21. B. Y. Chen, Jacobi's elliptic functions and Lagrangian immersions, Proc. Roy. Soc. Edinburgh Ser. A 126 (1996), 687-704.
22. B. Y. Chen, A general inequality for submanifolds in complex space forms and its applications, Arch. Math. (Basel) 67 (1996), 519-528.
23. B. Y. Chen, Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math. 99 (1997), 69-108.
24. B. Y. Chen, Complex extensors and Lagrangian submanifolds in complex space forms, Tôhoku Math. J. 49 (1997), 277-297.
25. B. Y. Chen, Strings of Riemannian invariants, inequalities, ideal immersions and their applications, in: The Third Pacific Rim Geometry Conference, Internat Press, Cambridge, MA, 1998, pp. 7-60.
26. B. Y. Chen, A vanishing theorem for Lagrangian immersions into Einstein-Kaehler manifolds and its applications, Soochow J. Math. 24 (1998), 157-164.
27. B. Y. Chen, Intrinsic and extrinsic structures of Lagrangian surfaces in complex space forms, Tsukuba J. Math. 22 (1998), 657-680.
28. B. Y. Chen, Representation of flat Lagrangian H-umbilical submanifolds in complex Euclidean spaces, Tôhoku Math. J. 50 (1999), 13-20.
29. B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. 41 (1999), 33-41.
30. B. Y. Chen, Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. 26 (2000), 105-127.
31. B. Y. Chen, Warped product, complex extensors and Lagrangian submanifolds, Soochow J. Math. 26 (2000), 1-17.
32. B. Y. Chen, On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms, Arch. Math. (Basel) 74 (2000), 154-160.
33. B. Y. Chen, Ideal Lagrangian immersions in complex space forms, Math. Proc. Cambridge Philos. Soc. 128 (2000), 511-533.
34. B. Y. Chen, Riemannian submanifolds, in: Handbook of Differential Geometry, vol. I, F. Dillen and L. Verstraelen, eds., North Holland Publ. 2000, pp. 187-418.
35. n satisfying a basic inequality, Arch. Math. (Basel) 63 (1994), 553-564.
36. B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Two equivariant totally real immersions into the nearly Kahler 6-sphere and their characterization, Japan. J. Math. 21 (1995), 207-222.
37. 6 by their sectional curvatures, Tôhoku Math. J. 47 (1995), 185-198.
38. 3 and its characterization, Proc. Roy. Soc. Edinburgh Ser. A 126 (1996), 153-165.
39. n(4c), Math. Proc. Cambridge Philos. Soc. 124 (1998), 107-125.
40. B. Y. Chen and C. S. Houh, Totally real submanifolds of a quaternion projective space, Ann. Mat. Pura Appl. 70 (1979), 185-199.
41. B. Y. Chen, C. S. Houh and H. S. Lue, Totally real submanifolds, J. Differential Geom. 12 (1977), 473-480.
42. 2 J. Math. Pures Appl. 66 (1987), 321-325.
43. B. Y. Chen and J. M. Morvan, Deformations of isotropic submanifolds in Kahler manifolds, J. Geom. Phys. 13 (1994), 79-104.
44. B. Y. Chen, J. M. Morvan and T. Nore, Energy, tension and finite type maps, Kodai Math. J. 9 (1986), 406-418.
45. B. Y. Chen and T. Nagano, Totally geodesic submanifolds of symmetric spaces, I Duke Math. J. 44 (1977), 745-755; -, II, ibid 45 (1978), 405-425.
46. B. Y. Chen and T. Nagano, Totally geodesic submanifolds of symmetric spaces, I Duke Math. J. 44 (1977), 745-755; -, II, ibid 45 (1978), 405-425.
47. B. Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257-266.
48. B. Y. Chen and K. Ogiue, Two theorems on Kaehler manifolds, Michigan Math. J. 21 (1974), 225-229.
49. B. Y. Chen and L. Vrancken, Lagrangian submanifolds satisfying a basic inequality, Math. Proc. Cambridge Philos. Soc. 120 (1996), 291-307.
50. B. Y. Chen and L. Vrancken, Lagrangian submanifolds of the complex hyperbolic space, to appear.
51. 1(4c), Tôhoku Math. J., to appear.
52. n and the symmetric generalized wave equation, Tôhoku Math. J. 47 (1995), 117-123.
53. 6(1) Glasgow Math. J. 40 (1998), 199-204.
54. 6 Nihonkai Math. J. 1 (1990), 193-201.
55. R. Deszcz, F. Dillen, L. Verstraelen and L. Vrancken, Quasi-Einstein totally real submanifolds of the nearly Kähler 6-sphere, Tôhoku Math. J. 51 (1999), 461-478.
56. F. Dillen, B. Opozda, L. Verstraelen and L. Vrancken, On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere, Proc. Amer. Math. Soc. 99 (1987), 741-749.
57. 6(1) with K ≥ 1/16, J. Math. Soc. Japan 42 (1990), 565-584.
58. 6(1) satisfying Chen's equality, Trans. Amer. Math. Soc. 348 (1996), 1633-1646.
59. I. Dimitric, 1-type submanifolds of the complex projective space, Kodai Math. J. 14 (1991), 281-295.
60. N. Ejiri, Totally real submanifolds in a 6-sphere, Proc. Amer. Math. Soc. 83 (1981), 759-763.
61. N. Ejiri, Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms, Proc. Amer. Math. Soc. 84 (1982), 243-246.
62. 4, Tokyo J. Math. 9 (1986), 297-324.
63. L. Gheysens, P. Verheyen and L. Verstraelen, Sur les surfaces A ou les surfaces de Chen, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 913-916.
64. M. L. Gromov, A topological technique for construction of solutions of differential equations and inequalities, in: Actes du Congress International des Mathematiciens, Tome 2, Gauthier-Villars, Paris, 1971, pp. 221-225.
65. M. L. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
66. R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47-157.
67. H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Verlag, Basel, 1994.
68. C. S. Houh and Y. Hong, Lagrangian submanifolds of quaternionic Kaehlerian manifolds satisfying Chen's equality, Beiträge Algebra. Geom. 39 (1998), 413-421.
69. U. H. Ki and Y. H. Kim, Totally real submanifolds of a complex space form, Internat. J. Math. Math. Sci. 19 (1996), 39-44.
70. M. Kotani, An immersion of an n-dimensional real space form into an n-dimensional complex space form, Tokyo J. Math. 9 (1986), 103-113.
71. M. Kriele and L. Vrancken, Minimal Lagrangian submanifolds of Lorentzian complex space forms with constant sectional curvature, Arch. Math. (Basel) 72 (1999), 223-232.
72. Le Khong Van and A. T. Fomenko, Lagrangian manifolds and the Maslov index in the theory of minimal surfaces, Soviet Math. Dokl. 37 (1988), 330-333.
73. Y. I. Lee, Lagrangian minimal surfaces in Kaehler-Einstein surfaces of negative scalar curvature, Comm. Anal. Geom. 2 (1994), 579-592.
74. H. Li, The Ricci curvature of totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere, Bull. Belg. Math. Soc. Simon Stevin 3 (1996), 193-199.
75. X. M. Liu, Totally real submanifolds in a complex projective space, Internat. J. Math. Math. Sci. 22 (1999), 205-208.
76. 2 that is not totally geodesic, Proc. Amer. Math. Soc. 53 (1975), 186-190.
77. K. Matsumoto, I. Mihai and A. Oiaga, Shape operator for slant submanifolds in complex space forms, Bull. Yamagata Univ. Natur. Sci. 14 (2000), 169-177.
78. 6, Tsukuba J. Math. 9 (1985), 185-202.
79. n(c), J. Ramanujan Math. Soc. 9 (1994), 13-24.
80. Y. Matsuyama, Curvature pinching for totally real submanifolds of a complex projective space, J. Math. Soc. Japan 52 (2000), 51-64.
81. C. Montealegre and L. Vrancken, Three dimensional Lagrangian submanifolds of the complex projective space, to appear.
82. S. Montiel, A. Ros and F. Urbano, Curvature pinching and eigenvalue rigidity for minimal submanifolds, Math. Z. 191 (1986), 537-548.
83. S. Montiel and F. Urbano, Isotropic totally real submanifolds, Math. Z. 199 (1988), 55-60.
84. J. M. Morvan, Classe de Maslov d'une immersion Lagranienne etminimalité, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 633-636.
85. H. Naitoh, Totally real parallel submanifolds, Tokyo J. Math. 4 (1981), 279-306.
86. H. Naitoh, Parallel submanifolds of complex space forms I, Nagoya Math. J. 90 (1983), 85-117;
87. II, ibid 91 (1983), 119-149.
88. . H. Naitoh and M. Takeuchi, Totally real submanifolds and symmetric bounded domain, Osaka J. Math. 19 (1982), 717-731.
89. K. Ogiue, K. Positively curved totally real minimal submanifolds immersed in a complex projective space, Proc. Amer. Math. Soc. 56 (1976), 264-266.
90. K. Ogiue, Some recent topics in the theory of submanifolds, Sugaku Expositions, 4 (1991), 21-41.
91. Y. G. Oh, Second variation and stability of minimal Lagrangian submanifolds in Kaehler manifolds, Invent. Math. 101 (1990), 501-519.
92. n(4c), Ph.D. Thesis, Michigan State University, 2000.
93. Y. Ohnita, Totally real submanifolds with nonnegative sectional curvature, Proc. Amer. Math. Soc. 97 (1986), 474-478.
94. B. Palmer, Buckling eigenvalues, Gauss maps and Lagrangian submanifolds, Differential Geom. Appl. 4 (1994), 391-403.
95. . B. Palmer, Calibrations and Lagrangian submanifolds, Tôhoku Math. J. 50 (1998), 303-315.
96. H. Reckziegel, Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion, in: Global Differential Geometry and Global Analysis 1984, Springer-Verlag, Berlin, Lecture Notes in Math., 1156, 1985, pp. 264-279.
97. A. Ros, Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J. 6 (1983), 88-99.
98. A. Ros and F. Urbano, Lagrangian surface in the complex Euclidean plane with conformal Maslov form, J. Math. Soc. Japan, 45 (1993, 565-582,
99. n with conformal Maslov form and the Whitney sphere, J. Math. Soc. Japan 50 (1998), 203-226.
100. B. Rouxel, Chen submanifolds, in: Geometry and Topology of Submanifolds, VI, World Scientific, River Edge, NJ, 1994, pp. 185-198.
101. C. U. Sánchez, A. L. Cali and J. L. Moreschi, Homogeneous totally real submanifolds of complex projective space, Rend Sem. Mat. Univ. Padova, 101 (1999), 83-94.
102. N. Sato, Totally real submanifolds of a complex space form with nonzero parallel mean curvature vector, Yokohama Math. J. 44 (1997), 1-4.
103. n, Chinese Sci. Bull. 40 (1995), 621-626.
104. R. Schoen and J. Wolfson, Minimizing volume among Lagrangian submanifolds, in: Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, RI, 1999. pp. 181-199.
105. B. Suceava, The Chen invariants of warped products of hyperbolic planes and their applications to immersibility problems, Tsukuba J. Math., to appear.
106. M. Takeuchi, Stability of certain minimal submanifolds of compact Hermitian symmetric spaces, Tôhoku Math. J. 36 (1984), 293-314.
107. F. Urbano, Totally real minimal submanifolds of a complex projective space, Proc. Amer. Math. Soc. 93 (1985), 332-334.
108. F. Urbano, Negatively curved totally real submanifolds, Math. Ann. 273 (1986), 345-348.
109. F. Urbano, Totally real submanifolds of a complex projective space, in: Geometry and Topology of Submanifolds, World Scientific, Teaneck, N.J., 1989, pp. 198-208.
110. n and the Laplacian of 1-forms, Geom. Dedicata 48 (1993), 309-318.
111. 6, Algebras Groups Geom. 5 (1988), 369-394.
112. 6, J. Math. Pure Appl. 77 (1998), 631-645.
113. N(1), Kodai Math. J. 15 (1992), 141-153.
114. S. T. Yau, Submanifolds with constant mean curvature, I, Amer. J. Math. 96 (1974), 346-366.
115. K. Yano, Totally real submanifolds of a Kaehlerian manifold, J. Differential Geom. 11 (1976), 351-359.
116. R. Aiyama, Totally real surfaces in the complex 2-space, preprint.