Journal ArticleDOI
A Penrose-like Inequality for the Mass of Riemannian Asymptotically Flat Manifolds
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In this article, an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature was shown.Abstract:
We prove an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature. Our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere (as in the original Penrose conjecture) and some nomalized Sobolev ratio. As expected, the equality case is achieved if and only if the metric is that of a standard spacelike slice in the Schwarzschild space.read more
Citations
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The inverse mean curvature flow and the Riemannian Penrose Inequality
Gerhard Huisken,Tom Ilmanen +1 more
TL;DR: In this article, a theory of weak solutions of the inverse mean curvature flow was developed and employed to prove the Riemannian Penrose inequality for each connected component of a 3-manifold of nonnegative scalar curvature.
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Proof of the riemannian penrose inequality using the positive mass theorem
TL;DR: The Riemannian Penrose Conjecture as mentioned in this paper is an important case of a con- jecture (41) made by Roger Penrose in 1973, by defining a new flow of metrics.
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The mass of asymptotically hyperbolic Riemannian manifolds
Piotr T. Chruściel,Marc Herzlich +1 more
TL;DR: In this article, a set of global invariants, called mass integrals, is defined for a large class of asymptotically hyperbolic Riemannian manifolds.
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Present status of the Penrose inequality
TL;DR: The Riemannian version of the Penrose inequality has been studied in the last decade or so as discussed by the authors, with a very interesting proposal to address the general case by Bray and Khuri.
Posted Content
The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature
TL;DR: In this paper, minimal surface techniques were used to prove the Penrose inequality in general relativity for two classes of 3-manifolds, and a new volume comparison theorem involving scalar curvature for 3-Manifolds followed from these same techniques.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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The Analysis of Linear Partial Differential Operators I
TL;DR: In this article, the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certain products using instruction manuals, which are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments.
Journal ArticleDOI
A new proof of the positive energy theorem
TL;DR: In this article, a new proof of the positive energy theorem of classical general relativity was given and it was shown that there are no asymptotically Euclidean gravitational instantons.
Journal ArticleDOI
On the proof of the positive mass conjecture in general relativity
Richard Schoen,Shing-Tung Yau +1 more
TL;DR: In this paper, it was shown that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat, which is the assumption of the existence of a maximal spacelike hypersurface.
Journal ArticleDOI
On the structure of manifolds with positive scalar curvature
Richard Schoen,Shing-Tung Yau +1 more
TL;DR: Schoen and Yau as mentioned in this paper showed that any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sn by a discrete subgroup of the conformal group.