Spacelike graphs with parallel mean curvature in pseudo-Riemannian product manifolds

Abstract

The author introduces the w-function defined on the considered spacelike graph M. Under the growth conditions w = o(log z) and w = o(r), two Bernstein type theorems for M in ℝ _m ^n+m are got, where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively. As the ambient space is a curved pseudo-Riemannian product of two Riemannian manifolds (Σ₁, g ₁) and (Σ₂, g ₂) of dimensions n and m, a Bernstein type result for n = 2 under some curvature conditions on Σ₁ and Σ₂ and the growth condition w = o(r) is also got. As more general cases, under some curvature conditions on the ambient space and the growth condition w = o(r) or $w = 0\left( {\sqrt r } \right)$, the author concludes that if M has parallel mean curvature, then M is maximal.

Keywords

Product manifold / Spacelike graph / Parallel mean curvature / Maximal / Bernstein

References

[1] Almgren F. J. Jr. Some interior regularity theorems for minimal surfaces and an extension of Bernsteins theorem. Ann. Math., 1966, 84: 277-292

[2] Bombieri E., de Giorgi E., Giusti E.. Minimal cones and the Bernstein problem. Invent. Math., 1969, 7: 243-268

[3] Calabi E.. Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Pure Math., 1970, 15: 223-230

[4] Chen Q., Xin Y. L.. A generalized maximal principle and its applications in geometry. Amer. J. Math., 1992, 114: 355-366

[5] Cheng S. Y., Yau S. T.. Differential equations on Riemannian manifolds and geometric applications. Comm. Pure Appl. Math., 1975, 28: 333-354

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

[7] Chern, S. S., On the Curvature of a Piece of Hypersurface in Euclidean Space, Abh. Math. Sem., Hamburg, 29, 1964.

[8] de Giorgi E.. Una estensione del teorema di Bernstein. Ann. Scuola Norm. Sup. Pisa., 1965, 19: 79-85

[9] Dong Y. X.. Bernstein theorems for space-like graphs with parallel mean curvature and controlled growth. J. Geom. Phys., 2008, 58: 324-333

[10] Flanders H.. Remark on mean curvature. J. London Math. Soc., 1966, 41: 364-366

[11] Jost J., Xin Y. L.. Some aspects of the global geometry of entire spacelike submaniflods. Results Math., 2001, 40: 233-245

[12] Jost, J., Xin, Y. L. and Yang, L., The Gauss image of entire graphs of higher codimension and Bernstein type theorems, Calc. Var. Part. Diff. Eq., to appear. arXiv: 1009.3901v1

[13] Li G. H., Salavessa I.. Graphic Bernstein results in curved pseudo-Riemannian manifolds. J. Geom. Phys., 2009, 59: 1306-1313

[14] Salavessa I.. Spacelike graphs with parallel mean curvature. Bull. Belg. Math. Soc., 2008, 15: 65-76

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

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

[17] Xin, Y. L., Minimal Submanifolds and Related Topics, World Scientific Publishing, Singapore, 2003.

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

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

[20] Xin Y. L.. Mean curvature flow with bounded Gauss image. Results Math., 2011, 59: 415-436

[21] Xin Y. L., Yang L.. Convex functions on Grassmannian manifolds and Lawson-Osserman problem. Adv. Math., 2008, 219(4): 1298-1326

[22] 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