Null hypersurfaces evolved by their mean curvature in a Lorentzian manifold

Abstract

We use null isometric immersions to introduce time-dependent null hypersurfaces, in a Lorentzian manifold, evolving in the direction of their mean curvature vector (a vector transversal to the null hypersurface). We prove an existence result for such hypersurfaces in a short time interval. Then, we discuss the evolution of some induced geometric objects. Consequently, we prove under certain geometric conditions that some of the above objects will blow-up in finite time. Also, several examples are given to illustrate the main ideas.