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 ℝmn+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, g1) and (Σ2, g2) 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.