Spacelike Submanifolds of Codimension Two with Parallel Mean Curvature Vector Field in Lorentz-Minkowski Spacetime Contained in the Light Cone

TL;DR

This paper establishes an integral inequality for compact spacelike codimension-two submanifolds with parallel mean curvature in Lorentz-Minkowski spacetime and characterizes those lying in the light cone as totally umbilical spheres.

Contribution

It introduces a new integral inequality and provides a complete classification of submanifolds with parallel mean curvature in the light cone.

Findings

Derived a general integral inequality for the submanifolds.

Proved a rigidity result characterizing submanifolds in the light cone.

Showed these submanifolds are totally umbilical spheres in a spacelike hyperplane.

Abstract

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to derive a rigidity result. Specifically, we obtain a complete characterization of all compact spacelike submanifolds with parallel mean curvature vector field that lie in the light cone of the Lorentz-Minkowski spacetime: they must be totally umbilical spheres contained in a spacelike hyperplane in Lorentz-Minkowski spacetime.