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The Einstein flow, the sigma-constant and the geometrization of 3-manifolds
Open accessAbstract
For the problem of the Hamiltonian reduction of Einstein's equations on a (3 + 1) vacuum spacetime that admits a foliation by constant mean curvature (CMC) compact spacelike hypersurfaces M that satisfy certain topological restrictions, we introduce a dimensionless non-local time-dependent reduced Hamiltonian system where and P reduced is the reduced phase space. For this system, we establish the following properties: (1) H reduced has a unique critical point at the hyperbolic point which is a strict local minimum; (2) H reduced is a strictly monotonically decreasing function along the Einstein flow unless , in which case H reduced is constant in time; and (3) the infimum of H reduced is related to the -constant of M by Further applications and developments relating the Einstein flow to ( M ) and to the geometrization conjecture for 3-manifolds are discussed.
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[12] proved a global existence theorem for expanding solutions in the same gauge by controlling the Dirichlet energy of an associated harmonic map for any τ in (-∞,0).
“In addition to the local well-posedness, [12] proved a global existence theorem for the expanding solutions in the same gauge by controlling the Dirichlet energy of an associated harmonic map for any τ ∈ ( -∞ , 0) .”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [12]
The Letter to the Editor describes τ ∈ (−∞, 0) as the expected maximal range, not a range established by a global-existence theorem. Section 5.1 discusses long-time existence as a potential future application. Searches for Dirichlet energy, harmonic maps, and a global-existence theorem found no passage establishing the attributed result. (LLM, conf=0.76)
Experimental — accuracy not yet measuredIn(τ) and S(τ) are continuous functions of τ by existence-uniqueness-continuity (well-posedness) of Einstein's equations.
“Both I n ( τ ) and S ( τ ) are continuous functions of τ by existence-uniqueness-continuity (or well-posedness) of the Einstein's equations [12].”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [12]
The Letter to the Editor defines a time-dependent Hamiltonian in terms of τ, and section 4 discusses its infimum, but neither passage identifies I_n(τ) or S(τ) or establishes their continuity. Searches for those functions and for well-posedness, existence, uniqueness, and continuous dependence found no passage making the attributed argument. (LLM, conf=0.74)
Experimental — accuracy not yet measured
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Research works printed but not linked: 7
- 10.1016/S0920-5632(97)00363-0 · DOI · Paper page
- 10.1063/1.528475 · DOI · Paper page
- Löbell F, 1931, Ber. Verhandl. Sächs. Akad. Wiss. Leipzig, Math. Phys. Kl., 83, 167
- Eardley D, 1986, Commun. Math. Phys., 106, 137, 10.1007/BF01210929 · DOI
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- 10.1090/S0273-0979-99-00773-9 · DOI
Cited by
Sorted by citations
Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space
Mondal, P. (2023) · Journal of Mathematical Physics · doi:10.1063/5.0136631 · Cited by 1
Metadata
- Authors
- Fischer, A. E., & Moncrief, V.
- Affiliations
- University of California, Santa Cruz (US)
- Yale University (US)
- ORCID
- Fischer, Arthur E
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1999-10-08
- Volume
- 16
- Issue
- 11
- Pages
- L79-L87
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2057406392
- Cited by
- 22
- References
- 0
- Citations by year
- 2025: 1 · 2023: 1 · 2015: 1
- FWCI
- 6.11
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- sigma constants, Hamiltonian reduction, vacuum Einstein equations, constant mean curvature hypersurfaces, spacelike hypersurfaces, geometrization conjecture, critical point analysis, hyperbolic geometry
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