Convex function on pseudo-Grassmann manifold and its applications for Bernstein-type theorem

Abstract

We derive the Hessian estimate of the ω-function defined on the pseudo-Grassmann manifold {\mathit{G}}_{n,m}^m, which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [Manuscripta Math., 2000, 103: 191-202]. We also estimate the squared norm of the second fundamental form of a complete spacelike submanifold in pseudo- Euclidean space in terms of the ω-function and the mean curvature.

Keywords

Pseudo-Euclidean space / pseudo-Grassmann manifold / Bernsteintype theorem

References

[1] Calabi E. Examples of Bernstein problems for some non-linear equations. Proc Sympos Pure Math, 1970, 15: 223-230

[2] Cheng S Y, Yau S T. Maximal spacelike hypersurfaces in the Lorentz-Minkowski space. Ann Math, 1976, 104: 407-419

[3] Hildebrandt S, Jost J, Widman K O. Harmonic mappings and minimal submanifolds. Invent Math, 1980, 62: 269-298

[4] Ishihara T. The harmonic Gauss maps in a generalized sense. J Lond Math Soc, 1982, 26: 104-112

[5] Ishihara T. Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature. Michigan Math J, 1988, 35: 345-352

[6] Omori H. Isometric immersions of Riemannian manifolds. J Math Soc Japan, 1967, 19: 205-214

[7] Triebergs A. Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent math, 1982, 66: 39-56

[8] Xin Y L. On Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comment Math Helv, 1991, 66: 590-598

[9] Xin Y L. Geometry of Harmonic Maps. Progr Nonlinear Differential Equations Appl, Vol 23. Boston: Birkhäuser, 1996

[10] Xin Y L. A rigidity theorem for a space-like graph of higher codimension. Manuscripta Math, 2000, 103: 191-202

[11] Xin Y L. Minimal Submanifolds and Related Topics. Singapore: World Scientific Publ, 2003

[12] Xin Y L, Yang L. Convex functions on Grassmannian manifolds and Lawson-Osserman problem. Adv Math, 2008, 219: 1298-1326

[13] Xin Y L, Ye R G. Bernstein-type theorems for space-like surfaces with parallel mean curvature. J Reine Angew Math, 1997, 489: 189-198

[14] Yau S T. Harmonic functions on complete Riemannian manifolds. Commun Pure Appl Math, 1975, 28: 201-208

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg