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On the Asymptotics for the Vacuum Einstein Constraint Equations
Abstract
In this paper we prove density of asymptotically flat solutions with special asymptotics in general classes of solutions of the vacuum constraint equations. The first type of special asymptotic form we consider is called harmonic asymptotics. This generalizes in a natural way the conformally flat asymptotics for the K = 0 constraint equations. We show that solutions with harmonic asymptotics form a dense subset (in a suitable weighted Sobolev topology) of the full set of solutions. An important feature of this construction is that the approximation allows large changes in the angular momentum. The second density theorem we prove allows us to approximate asymptotically flat initial data on a three-manifold M for the vacuum Einstein field equation by solutions which agree with the original data inside a given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The construction generalizes work in [C], where the time-symmetric (K = 0) case was studied.
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“There exists a subspace W of pairs (u, Y ) ∈ W 2,p -q and a finite dimensional subspace K ⊂ W 2,p -q × W 1,p -1-q of pairs (h, w) ∈ C ∞ c such that T : W × K -→ L p -2-q is a diffeomorphism from a neighborhood of 0 in W ×K onto a ball centered at Φ (g,π) (g, π) in L p -2-q .”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [8]
In the citing paper
“The modified constraint map Φ (g,π) : M 2,p -q ×W 1,p -1-q -→ L p -2-q is smooth, and DΦ (g,π) : W 2,p -q × W 1,p -1-q -→ L p -2-q is surjective.”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [8]
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Cited by
Quasi-Local Energy-Momentum and Angular Momentum in General Relativity
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Lessons from the Information Paradox
Raju, S. (2020) · arXiv · doi:10.48550/arxiv.2012.05770 · Cited by 234
The spacetime positive mass theorem in dimensions less than eight
Eichmair, M., Huang, L.-H., Lee, D. A., & Schoen, R. (2015) · Journal of the European Mathematical Society · doi:10.4171/jems/584 · Cited by 82
Self-citationEquality in the Spacetime Positive Mass Theorem
Huang, L.-H., & Lee, D. A. (2020) · Communications in Mathematical Physics · doi:10.1007/s00220-019-03619-w · Cited by 32
Metadata
- Authors
- Corvino, J., & Schoen, R. M.
- Venue
- Journal of Differential Geometry
- Publisher
- International Press of Boston
- Published
- 2006-06-01
- Volume
- 73
- Issue
- 2
- Pages
- 185-217
- ISSN
- 0022-040X
- Cited by
- 143
- References
- 0
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