The gap phenomenon for conformally related Einstein metrics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910290089607168 |
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| author | Gregorovič, Jan Šilhan, Josef |
| author_facet | Gregorovič, Jan Šilhan, Josef |
| contents | We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension $n$ of the conformally nonflat conformal manifold. In the Riemannian case, these two dimensions are at most $n-3$ and $\frac{(n-4)(n-3)}{2}$, respectively. In the Lorentzian case, these two dimensions are at most $n-2$ and $\frac{(n-3)(n-2)}{2}$, respectively. In the remaining signatures, these two dimensions are at most $n-1$ and $\frac{(n-2)(n-1)}{2}$, respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)-Euclidean base of dimension $n-4$ with one of the 4-dimensional submaximal examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The gap phenomenon for conformally related Einstein metrics Gregorovič, Jan Šilhan, Josef Differential Geometry 58J70, 53C18, 53C25, 58J60 We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension $n$ of the conformally nonflat conformal manifold. In the Riemannian case, these two dimensions are at most $n-3$ and $\frac{(n-4)(n-3)}{2}$, respectively. In the Lorentzian case, these two dimensions are at most $n-2$ and $\frac{(n-3)(n-2)}{2}$, respectively. In the remaining signatures, these two dimensions are at most $n-1$ and $\frac{(n-2)(n-1)}{2}$, respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)-Euclidean base of dimension $n-4$ with one of the 4-dimensional submaximal examples. |
| title | The gap phenomenon for conformally related Einstein metrics |
| topic | Differential Geometry 58J70, 53C18, 53C25, 58J60 |
| url | https://arxiv.org/abs/2401.03909 |