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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2204.03607 |
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| _version_ | 1866914306905341952 |
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| author | Avalos, Rodrigo Laurain, Paul Marque, Nicolas |
| author_facet | Avalos, Rodrigo Laurain, Paul Marque, Nicolas |
| contents | In this paper we prove some rigidity theorems associated to $Q$-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by $J_g$. This allows us to prove that Yamabe positive $J$-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this $J$-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of $J$ as a fourth order analogue to the Ricci tensor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_03607 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces Avalos, Rodrigo Laurain, Paul Marque, Nicolas Differential Geometry Mathematical Physics 53C24 (Primary) 83D05, 53C21 (Secondary) In this paper we prove some rigidity theorems associated to $Q$-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by $J_g$. This allows us to prove that Yamabe positive $J$-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this $J$-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of $J$ as a fourth order analogue to the Ricci tensor. |
| title | Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces |
| topic | Differential Geometry Mathematical Physics 53C24 (Primary) 83D05, 53C21 (Secondary) |
| url | https://arxiv.org/abs/2204.03607 |