ϕ\phi ϕ -Parabolicity and the Uniqueness of Spacelike Hypersurfaces Immersed in a Spatially Weighted GRW Spacetime

In this paper, we extend a technique due to Romero et al. establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime whose fiber has phi-parabolic universal Riemannian covering. As some applications of this criteria, we obtain uniqueness results concerning spacelike hypersurfaces immersed in spatially weighted generalized Robertson–Walker spacetimes. Furthermore, Calabi–Bernstein type results are also given

Height and Area Estimates for Constant Mean Curvature Graphs in E(κ,τ)\mathbb {E}(\kappa ,\tau ) E ( κ , τ ) -Spaces

We obtain area growth estimates for constant mean curvature graphs in E(k,t)- spaces with k<=0, by finding sharp upper bounds for the volume of geodesic balls in E(k,t). We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in E(k,t) with critical mean curvature have at most cubic intrinsic area growth. We also obtain sharp upper bounds for the extrinsic area growth of graphs with zero boundary values, and study distinguished examples in detail such as invariant surfaces, k-noids and ideal Scherk graphs. Finally we give a relation between height and area growth of minimal graphs in the Heisenberg space ( k= 0), and prove a Collin-Krust type estimate for such minimal graphs. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdA