Desigualdade de Caffarelli-Kohn-Nirenberg e solitons de Yamabe gradiente
This thesis deals with two distinct problems. Namely, we study [(P1)] Rigidity of metric spaces that support CKN inequality; [(P2)] Gradient Yamabe solitons on top of warped product manifolds B x f F. For the first problem, we prove that the metric measure spaces that support the CKN inequality have...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2019 |
| País: | Brasil |
| Institución: | Universidade Federal de Goiás (UFG) |
| Repositorio: | Repositório Institucional da UFG |
| Idioma: | portugués |
| OAI Identifier: | oai:repositorio.bc.ufg.br:tede/9718 |
| Acceso en línea: | http://repositorio.bc.ufg.br/tede/handle/tede/9718 |
| Access Level: | acceso abierto |
| Palabra clave: | Desigualdade de Caffarelli-Kohn-Nirenberg (CKN) Rigidez Solitons de Yamabe gradiente Produto torcido Curvatura escalar Estimativa de gradiente Caffarelli-Kohn-Nirenberg inequality Rigidity Gradient Yamabe solitons Warped product Scalar curvature Gradient estimates CIENCIAS EXATAS E DA TERRA::MATEMATICA |
| Sumario: | This thesis deals with two distinct problems. Namely, we study [(P1)] Rigidity of metric spaces that support CKN inequality; [(P2)] Gradient Yamabe solitons on top of warped product manifolds B x f F. For the first problem, we prove that the metric measure spaces that support the CKN inequality have n-dimensional volume growth, that is, there exists a universal constant C 0gt; 0 such that, m(B x (ρ)) ≥ C 0 ρ n , ∀x ∈ M, ρ gt; 0. As application, some rigidity theorems are obtained in the following spaces: Riemannian manifolds, Finsler manifolds and Alexandrov spaces. For the second problem, taking a gradient Yamabe soliton (B x f F, g, h, ρ), we obtain triviality results for h and f by means of some hypotheses on the base B. Furthermore, under a hypothesis involving the Ricci curvature of the base Ric gB , we prove estimates for h, f and for scalar curvature scal g , in addition, by means of a warping gradient estimates, we present a beautiful obstruction in the construction of gradient Yamabe solitons on warped product manifolds. Finally, by making use of invariant solution techniques, we classify all steady gradient Yamabe solitons with a conformally flat base that is invariant by the action of a codimension 1 translation group. |
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