Resumen de Spacelike mean curvature flow solitons, polynomial volume growth and stochastic completeness of spacelike hypersurfaces immersed into pp-vave spacetimes
Marco A. L. Velásquez, Henrique F. de Lima, José H. H. de Lacerda
Our purpose in this paper is to study some geometric properties of spacelike hypersurfaces immersed into a pp-wave spacetime, namely, a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold, we obtain sufficient conditions which guarantee that a complete noncompact spacelike hypersurface with polynomial volume growth is either totally geodesic, maximal or 1-maximal. As a consequence, we establish nonexistence results concerning such spacelike hypersurfaces. Next, using a weak form of the Omori–Yau maximum principle, we get uniqueness and nonexistence results for stochastically complete spacelike hypersurface with constant mean curvature. Finally, we establish the notion of spacelike mean curvature flow soliton in pp-wave spacetimes and we provide some geometric conditions that allow us to guarantee how close a complete spacelike mean curvature flow soliton is to a totally geodesic immersion.
