Compact spacelike submanifolds in the light cone with parallel mean curvature are exactly round totally umbilical spheres in spacelike hyperplanes.
Brinkmann, On Riemannian spaces conformal to Euclidean space, Proc
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Spacelike Submanifolds of Codimension Two with Parallel Mean Curvature Vector Field in Lorentz-Minkowski Spacetime Contained in the Light Cone
Compact spacelike submanifolds in the light cone with parallel mean curvature are exactly round totally umbilical spheres in spacelike hyperplanes.