Geometry of Normal Graphs in Euclidean Space and Applications to the Penrose Inequality in Minkowski

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 [1] 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.

A toy Penrose inequality and its proof

“…Since we are dealing with a geometric property of surfaces, one may wonder if inequalities similar to (12) hold true in other contexts. Given the difficulty to prove it in general, what if we were to lower the dimension?…”

“…Lately progress has been made in the original setup [11,12]. The original idea has also been generalized to other spacetime geometries-including Minkowski as a special case-for instance Schwarzschild spacetime [13], and asymptotically flat spacetimes satisfying the dominant energy condition in general [14].…”

The null Penrose inequality and the shell version in Minkowski

“…This is also discussed in full detail in Chapter 3, which we conclude by giving an alternative proof of Brendle and Wang's main result [15] mentioned before. In fact our proof of this result was simultaneous and independent of Brendle and Wang's, and used the projection identities described in Chapter 3, which were published in [75].…”

“…In the following theorem we quote Brendle and Wang result [15] and give a proof which in fact was obtained independently before their result was published. Our proof uses the machinery developed at the beginning of the chapter (and published in [75]), which allows us to express the shell Penrose inequality for the initial surface S in Minkowski in terms of the projected geometry. Let S be a closed, connected, orientable and spacetime convex surface in (M 1,n+1 , η) with contravariant metric γ −1 .…”

“…The computations are performed in the static setup but, for the sake of generality we also give the expressions in the case when the ambient manifold is of euclidean signature (and admits a hypersurface orthogonal nowhere zero Killing vector). The results in this chapter have been published in [75].…”