Specifically, if (M, g) is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass m, and A is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime.
This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 in the case where A is the area of the largest component of the outermost minimal surface.
The original physical argument that led Penrose to conjecture such an inequality invoked the Hawking area theorem and the cosmic censorship hypothesis.
More generally, Penrose conjectured that an inequality as above should hold for spacelike submanifolds of spacetimes that are not necessarily time-symmetric.
Proving such an inequality remains an open problem in general relativity, called the Penrose conjecture.