Constant mean curvature solutions of the Einstein constraint equations on closed manifolds
Abstract
We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.
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The Dirichlet energy of the identity harmonic map is a proper function on Teichmüller space.
“Note that the conformal and diffeomorphism invariance of E [ id ; g, ρ ] allow it to be a function on the Teichmüller space of Σ p and in particular a proper function (i.e., the inverse images of the compact sets are compact)[19, 35].”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [35]
There exists a harmonic map homotopic to the identity, and such a map is an orientation-preserving diffeomorphism.
“From [34, 35], we know that there is a harmonic map homotopic to the identity i.e., E ∈ D 0 and in fact such a map is an orientation preserving diffeomorphism.”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [35]
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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 · see citing works
- Authors
- Isenberg, J.
- Affiliations
- University of Oregon (US)
- Corresponding author
- Isenberg, James
- ORCID
- Isenberg, James
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1995-09-01
- Volume
- 12
- Issue
- 9
- Pages
- 2249-2274
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W1994815733
- Cited by
- 218
- References
- 0
- Citations by year
- 2026: 3 · 2025: 5 · 2024: 9 · 2023: 3 · 2022: 11 · 2021: 4 · 2020: 3 · 2019: 3 · 2018: 3 · 2017: 8 · 2016: 12 · 2015: 11 · 2014: 19 · 2013: 5 · 2012: 5
- FWCI
- 4.16
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Classical and Quantum Gravity (publishedVersion)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- Einstein constraint equations, constant mean curvature, closed manifolds, parametrization of solutions, general relativity
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