Existence of outermost apparent horizons with product of spheres topology

@article{Schwartz2007ExistenceOO,
  title={Existence of outermost apparent horizons with product of spheres topology},
  author={Fernando Schwartz},
  journal={arXiv: General Relativity and Quantum Cosmology},
  year={2007}
}
  • F. Schwartz
  • Published 18 April 2007
  • Mathematics
  • arXiv: General Relativity and Quantum Cosmology
In this paper we find new examples of Riemannian manifolds with outermost apparent horizons with nonspherical topology, in dimensions four and above. More precisely, for any $n,m\ge1$, we construct asymptotically flat, scalar flat Riemannian manifolds containing smooth outermost minimal hypersurfaces with topology $S^n\times S^{m+1}$. In the context of general relativity these hypersurfaces correspond to outermost apparent horizons of black holes. 
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References

SHOWING 1-10 OF 58 REFERENCES
The inverse mean curvature flow and the Riemannian Penrose Inequality
Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m…
PROOF OF THE RIEMANNIAN PENROSE INEQUALITY USING THE POSITIVE MASS THEOREM
We prove the Riemannian Penrose Conjecture, an important case of a con- jecture (41) made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of…
Rigidity of outer horizons and the topology of black holes
In a recent paper [13] the author and Rick Schoen obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. It was proved that, apart…
The mass of an asymptotically flat manifold
We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic…
A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions
Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy…
A rotating black ring solution in five dimensions.
The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S1xS2. It…
Regularity Theory for Mean Curvature Flow
1 Introduction.- 2 Special Solutions and Global Behaviour.- 3 Local Estimates via the Maximum Principle.- 4 Integral Estimates and Monotonicity Formulas.- 5 Regularity Theory at the First Singular…
On the structure of manifolds with positive scalar curvature
Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are…
Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature
Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some…
Stable and singular solutions of the equation Δu = 1/u
hhWe study properties of the semilinear elliptic equation Δu = 1/u on domains in R n , with an eye toward non-negative singular solutions as limits of positive smooth solutions. We prove the…
...
1
2
3
4
5
...