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- Hubert L. Bray
- 2002

We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold… (More)

In this paper we compute the σ-invariants (sometimes also called the smooth Yamabe invariants) of RP and RP × S1 (which are equal) and show that the only prime 3-manifolds with larger σ-invariants are S3, S2 × S1, and S2×̃S1 (the nonorientable S2 bundle over S1). More generally, we show that any 3-manifold with σ-invariant greater than RP is either S3, a… (More)

In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques. The Penrose inequality in general relativity is closely related to the… (More)

In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a… (More)

We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominant energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data,… (More)

- Hubert L. Bray, Dan A. Lee
- 2009

The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the… (More)

- Hubert L. Bray
- 2010

We define geometric axioms for the metric and the connection of a spacetime where the gravitational influence of the connection may be interpreted as dark matter. We show how these axioms lead to the Einstein-Klein-Gordon equations with a cosmological constant, where the scalar field of the Klein-Gordon equation represents the deviation of the connection… (More)

Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered… (More)

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such “zero area singularities” in Riemannian manifolds, generalizing far beyond the Schwarzschild case… (More)

In 1973, R. Penrose presented an argument that the total mass of a space-time which contains black holes with event horizons of total area A should be at least √ A/16π. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. This… (More)