Areas and volumes for null cones

Understand

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold.

• We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volume monotonicity theorem and Guenther's volume comparison theorem.
• We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above.