Hubert L. Bray

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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)
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)