Spacelike mean curvature flow solitons, polynomial volume growth and stochastic completeness of spacelike hypersurfaces immersed into pp-wave spacetimes (Q6141132)

The pp-wave spacetimes belong to the class of exact solutions of Einstein's field equation admitting a nonexpanding, shear-free and twist-free null congruence, which can model radiation (electromagnetic or gravitational) moving at the speed of light. The aim of this manuscript is to carry out a study of geometric properties of complete space-like hypersurfaces \(\Sigma\) immersed in a pp-wave spacetime \(M^{n+1}\), which is endowed with a parallel light-like vector field \(\xi\). The authors obtain sufficient conditions which guarantee that a complete noncompact space-like hypersurface with polynomial volume growth is either totally geodesic, maximal or \(1\)-maximal. As a consequence, they establish nonexistence results concerning such space-like hypersurfaces. Moreover, they get uniqueness and nonexistence results for stochastically complete space-like hypersurface with constant mean curvature.