Geometry of normal graphs in Euclidean space and applications to the Penrose inequality in Minkowski

TL;DR

This paper explores the geometry of normal graphs in Euclidean space and applies these insights to analyze the Penrose inequality in Minkowski spacetime, linking intrinsic and extrinsic geometries through projections and height functions.

Contribution

It provides explicit relations between the geometry of hypersurfaces as normal graphs and the conditions for the Penrose inequality in Minkowski space.

Findings

Derived explicit conditions for Penrose inequality using height functions.

Connected hypersurface geometry with projections along null cones.

Included new results on projections in static spacetimes.

Abstract

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces S in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of S onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang's condition explicitly in terms of the time height function of S over a hyperplane and the geometry of the projection of S along its past null cone onto this hyperplane. We also include, in an Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime.