Complete Vertical Graphs with Constant Mean Curvature in Semi-Riemannian Warped Products

Abstract

In this paper we study complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. We first derive suitable formulas for the Laplacians of the height function and of a support-like function naturally attached to the graph; then, under appropriate restrictions on the values of the mean curvature and the growth of the height function, we obtain necessary conditions for the existence of such a graph. In the two-dimensional case we apply this analytical framework to state and prove Bernstein-type results in each of these ambient spaces.

On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

“…Suppose that = 2, then we have ( − 1) ≥ 2 − 1 for all ≥ 1 if ≥ 1 + 2/( − 1) > 0. Thus, our Theorem 9 extends Theorem 4.5 in [19]. At last, we write the uniqueness theorems for surface in steady state space which follows from Theorems 1 and 9, respectively, as following.…”

“…Recently, some uniqueness theorems for steady state space were obtained by [13,19]. In fact, Caminha and de Lima [19] proved the following results.…”

A Bernstein type theorem in ℝ × ℍ n

“…In addition, as a particular case of the Proposition 3.1 of [4] (see also Lemma 4.2 of [5]), we obtain the Laplacian on n of the angle function η. Lemma 2.1. Let ψ : n → R × M n be a hypersurface with orientation N , and let η = N , ∂ t be its angle function.…”

On the Umbilicity of Hypersurfaces in the Hyperbolic Space

“…In [20], Shu proved that a complete hypersurface in H n+1 with constant normalized scalar curvature and nonnegative sectional curvature must be either totally umbilical or isometric to a hyperbolic cylinder of H n+1 . Next, the third author and Caminha [11] studied complete vertical graphs of constant mean curvature in H n+1 . Under appropriate restrictions on the values of the mean curvature and the growth of the height function, they established necessary conditions for the existence of such a graph Σ n and, when n = 2, they proved that Σ 2 must be a horosphere.…”