The Dirichlet-to-Neumann Map for Poincaré-Einstein Fillings
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917037882736640 |
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| author | Blitz, Samuel Gover, A. Rod Kopiński, Jarosław Waldron, Andrew |
| author_facet | Blitz, Samuel Gover, A. Rod Kopiński, Jarosław Waldron, Andrew |
| contents | We study the non-linear Dirichlet-to-Neumann map for the Poincaré-Einstein filling problem. For even dimensional manifolds the range of this non-local map is described in terms of a rank two "Dirichlet-to Neumann tensor" along the boundary determined by the Poincaré-Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincaré-Einstein metrics. We construct natural "Dirichlet-to-Neumann hypersurface invariants" that are conformally invariant and recover all Dirichlet-to-Neumann tensors. We give an explicit formula for these hypersurface invariants and use a new vanishing result for odd order $T$-curvatures to show that they are the unique, natural conformal hypersurface invariant of transverse order equaling the boundary dimension. We also construct such conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings for odd dimensional manifolds with conformally flat boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08470 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Dirichlet-to-Neumann Map for Poincaré-Einstein Fillings Blitz, Samuel Gover, A. Rod Kopiński, Jarosław Waldron, Andrew Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics 53C18, 53A55, 53C21, 58J32 We study the non-linear Dirichlet-to-Neumann map for the Poincaré-Einstein filling problem. For even dimensional manifolds the range of this non-local map is described in terms of a rank two "Dirichlet-to Neumann tensor" along the boundary determined by the Poincaré-Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincaré-Einstein metrics. We construct natural "Dirichlet-to-Neumann hypersurface invariants" that are conformally invariant and recover all Dirichlet-to-Neumann tensors. We give an explicit formula for these hypersurface invariants and use a new vanishing result for odd order $T$-curvatures to show that they are the unique, natural conformal hypersurface invariant of transverse order equaling the boundary dimension. We also construct such conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings for odd dimensional manifolds with conformally flat boundary. |
| title | The Dirichlet-to-Neumann Map for Poincaré-Einstein Fillings |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics 53C18, 53A55, 53C21, 58J32 |
| url | https://arxiv.org/abs/2307.08470 |