Convex functions and spacetime geometry (Q2769509)

The authors show how the existence of a convex function on a spacetime places restrictions on the properties of the spacetime geometry. They consider convex functions on Riemannian manifolds which are regarded as spacelike submanifolds embedded in a spacetime or as Euclidean solutions of Einstein's equation. The existence of convex functions of strictly or uniformly convex type is incompatible with the existence of closed minimal submanifolds. The relation between the convex functions and Killing vector fields is discussed. Two definitions for convex functions on Lorentzian manifolds are given: a classical convex function and a spacetime convex function. The connection between spacetime convex functions and homothetic Killing vector fields is discussed. It is shown that, if a spacetime admits a spacetime convex function, then such a spacetime cannot have a marginally inner and outer trapped surface. The case when a cosmological spacetime admits a spacetime convex function is considered. The existence of a spacetime convex function on specific spacetimes, de Sitter, anti-de Sitter and black hole spacetimes, is also examined. The level sets of convex functions and foliations are discussed. Then, constant mean curvature foliations, barriers and convex functions are considered.