Lagrangian submanifolds in complex projective space CPⁿ

Abstract

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CPⁿ. Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CPⁿ.

Keywords

Lagrangian submanifold / second fundamental form / Maurer-Cartan form

References

[1] Castro I., Li H. Z., Urbano F. Hamiltonian-minimal Lagrangian submanifolds in complex space forms. Pacific J Math, 2006, 227(1): 43-63

[2] Chen B. Y., Dillen F., Verstraelen L., Vrancken L. An exotic totally real minimal immersion of S³ in CP³ and its characterisation. Proc Roy Soc Edinburgh Sect A, 1996, 126: 153-165

[3] Chen Q., Xu S. L. Rigidity of compact minimal submanifolds in a unit sphere. Geom Dedicata, 1993, 45: 83-88

[4] Chern S. S., do Carmo M., Kobayashi S. Browder F. E. Minimal submanifolds of a sphere with second fundamental form of constant length. Functional Analysis and Related Fields, 1970, Berlin: Springer, 59-75

[5] Ejiri N. Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms. Proc Amer Math Soc, 1982, 53: 186-190

[6] Ge J. Q., Tang Z. Z. A proof of the DDVV conjecture and its equality case. Pacific J Math, 2008, 273(1): 87-95

[7] Griffiths P. On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence question in differential geometry. Duke Math J, 1974, 41: 775-814

[8] Li A. M., Li J. M. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch Math, 1992, 58: 582-594

[9] Li A. M., Zhao G. S. Totally real minimal submanifolds in CPⁿ. Arch Math, 1994, 62: 562-568

[10] Ma H. Hamiltonian stationary Lagrangian surfaces in CP². Ann Global Anal Geom, 2005, 27: 1-16

[11] Ma H., Schmies M. Examples of Hamiltonian stationary Lagrangian Tori in CP². Geom Dedicata, 2006, 118: 173-183

[12] Onihta Y. Totally real submanifolds with non-negative sectional curvature. Proc Amer Math Soc, 1986, 97: 171-178

[13] Simons J. Minimal varieties in Riemannian manifolds. Ann Math, 1968, 88: 62-105

[14] Singley D. H. Smoothness theorems for the principal curvatures and principal vectors of a hypersurface. Rocky Mountain J Math, 1975, 5(1): 135-144

[15] Yang K. Complete and Compact Minimal Surfaces, 1989, Dordrecht: Kluwer Academic Publishers