Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915130295451648 |
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| author | Gomes, José Nazareno Vieira Tokura, Willian Isao |
| author_facet | Gomes, José Nazareno Vieira Tokura, Willian Isao |
| contents | We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry-Émery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE Gomes, José Nazareno Vieira Tokura, Willian Isao Differential Geometry We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry-Émery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results. |
| title | Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2407.10337 |