OtherPerspectives in Scalar Curvature · February 1, 2023
In the pioneering work of Stern, level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well ...
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Journal ArticlePure and Applied Mathematics Quarterly · January 1, 2019
Flatly Foliated Relativity (FFR) is a new theory which conceptually lies between Special Relativity (SR) and General Relativity (GR), in which spacetime is foliated by flat Euclidean spaces. While GR is based on the idea that “matter curves spacetime”, FFR ...
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Journal ArticleAnnales Henri Poincare · June 1, 2016
In this sequel paper, we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof whic ...
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Journal ArticleArchiv Der Mathematik · June 29, 2015
We provide new results and new proofs of results about the torsion of curves in $${\mathbb{R}^3}$$R3. Let $${\gamma}$$γ be a smooth curve in $${\mathbb{R}^3}$$R3 that is the graph over a simple closed curve in $${\mathbb{R}^2}$$R ...
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Journal ArticleCommunications in Mathematical Physics · April 1, 2015
We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show th ...
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Journal ArticleAsian Journal of Mathematics · 2013
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 ...
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Journal ArticleAsian Journal of Mathematics · January 1, 2011
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 existenc ...
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Journal ArticleCommunications on Pure and Applied Mathematics · September 1, 2010
Let (M, g) be a compact Riemannian manifold of dimension 3, and let F denote the collection of all embedded surfaces homeomorphic to R{double-struck}P{double-struck}2. We study the infimum of the areas of all surfaces in F . This quantity is rel ...
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Journal ArticleDiscrete and Continuous Dynamical Systems · June 1, 2010
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 exist ...
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Journal ArticleCommunications in Analysis and Geometry · January 1, 2010
We give a sharp upper bound for the area of a minimal two-sphere in a three-manifold (M,g) with positive scalar curvature. If equality holds, we show that the universal cover of (M,g) is isometric to a cylinder. ...
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Journal ArticleDuke Mathematical Journal · May 1, 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 f ...
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Journal ArticleInventiones Mathematicae · June 1, 2008
Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is de ...
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Journal ArticleCommunications in Mathematical Physics · May 1, 2007
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 mo ...
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Chapter · January 1, 2004
In a paper, R Penrose (1973) made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area A should be at least. ...
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Journal ArticleAnnals of Mathematics · January 2004
In this paper we compute the σ-invariants (sometimes also called the smooth Yamabe invariants) of RP3 and RP2×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 ...
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Journal ArticleCommunications in Analysis and Geometry · January 1, 2002
The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (ℝ3 δij). In this paper, we quantify this statement using spinors ...
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ConferenceProceedings of the American Mathematical Society · January 1, 2002
We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric (n - 1)-spheres of a spherically symmetric n-manifold are isoperimetric hypersurfaces, meaning that they min ...
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Journal ArticleJournal of Differential Geometry · January 1, 2001
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 ...
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Journal ArticleBio Systems · January 1995
The Shannon sampling theorem asserts that a continuous square-integrable function on the real line which has a compactly supported Fourier transform is uniquely determined by its restriction to a uniform lattice of points whose density is determined by the ...
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