Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner–Nordström spacetimes

Abstract

For each spacetime of a family of static spacetimes, we prove the existence of entire spherically symmetric spacelike graphs with prescribed mean curvature function. In particular, classical Schwarzschild and Reissner–Nordström spacetimes are considered. In both cases, the entire spacelike graph asymptotically approaches the event horizon. Spacelike graphs of constant mean curvature remain as a particular situation in the existence results, obtaining explicit expressions for the solutions. The proof of the results is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball, together with the obtention of a suitable bound for the length of the gradient of a solution which permits the prolongability to the whole space.