A large data result for vacuum Einstein's equations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908935219314688 |
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| author | Mondal, Puskar |
| author_facet | Mondal, Puskar |
| contents | We prove a global well-posedness and asymptotic convergence theorem for the \((3+1)\)-dimensional vacuum Einstein equations with positive cosmological constant \(Λ\) on globally hyperbolic spacetimes \(\widetilde M \cong M \times \mathbb R\), where \(M\) is a closed three-manifold of negative Yamabe type. In constant-mean-curvature transported spatial coordinates, an open set of large initial data gives rise to future-global solutions whose renormalized spatial metrics converge smoothly to a limiting metric of constant negative scalar curvature. The key new ingredient is an integrable damping mechanism, induced by the cosmological constant in this gauge and absent in the \(Λ=0\) vacuum problem, which yields time-integrable decay for the nonlinear evolution. As a consequence, the Einstein--\(Λ\) flow does not in general canonically encode the Thurston geometrization of the underlying three-manifold. This confirms a conjecture of Ringström on the asymptotic topological indistinguishability of large-data Einstein--\(Λ\) dynamics. An analogous theorem is also proved for manifolds of positive Yamabe type, under an additional technical hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A large data result for vacuum Einstein's equations Mondal, Puskar General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry We prove a global well-posedness and asymptotic convergence theorem for the \((3+1)\)-dimensional vacuum Einstein equations with positive cosmological constant \(Λ\) on globally hyperbolic spacetimes \(\widetilde M \cong M \times \mathbb R\), where \(M\) is a closed three-manifold of negative Yamabe type. In constant-mean-curvature transported spatial coordinates, an open set of large initial data gives rise to future-global solutions whose renormalized spatial metrics converge smoothly to a limiting metric of constant negative scalar curvature. The key new ingredient is an integrable damping mechanism, induced by the cosmological constant in this gauge and absent in the \(Λ=0\) vacuum problem, which yields time-integrable decay for the nonlinear evolution. As a consequence, the Einstein--\(Λ\) flow does not in general canonically encode the Thurston geometrization of the underlying three-manifold. This confirms a conjecture of Ringström on the asymptotic topological indistinguishability of large-data Einstein--\(Λ\) dynamics. An analogous theorem is also proved for manifolds of positive Yamabe type, under an additional technical hypothesis. |
| title | A large data result for vacuum Einstein's equations |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2502.11289 |