Uma caracterização espectral para os H(r)-toros na esfera
In this thesis we obtain some spectral estimates to characterize the Clifford hypersurfaces or H(r)-torus in the sphere S^n+1. The work was divided into two parts. In the first part we consider hypersurfaces closed in S^n+p with p ≥ 1. Initially, we proved that the only surfaces that maximize the se...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2020 |
| País: | Brasil |
| Institución: | Universidade Federal de Minas Gerais (UFMG) |
| Repositorio: | Repositório Institucional da UFMG |
| Idioma: | portugués |
| OAI Identifier: | oai:repositorio.ufmg.br:1843/44583 |
| Acceso en línea: | http://hdl.handle.net/1843/44583 |
| Access Level: | acceso abierto |
| Palabra clave: | Superfície de curvatura média constante Índice de Morse Estabilidade Autovalor forte e fraco Operador de Jacobi H(r)-toros Matemática – Teses Superficies de curvatura constante – Teses Autovalores – Teses Morse, Teoria de – Teses Jacobi, Metodos de – Teses |
| Sumario: | In this thesis we obtain some spectral estimates to characterize the Clifford hypersurfaces or H(r)-torus in the sphere S^n+1. The work was divided into two parts. In the first part we consider hypersurfaces closed in S^n+p with p ≥ 1. Initially, we proved that the only surfaces that maximize the second strong eigenvalue of the Jacobi operator in S^p+2 are the minimal Clifford torus, for this we use a technique based on the use of conformal applications. Then we use the same technique to prove that the estimate is true for the general case, assuming a hypothesis about the scalar curvature. Finishing the first part, we study a conjecture of classification of hypersurfaces not totally geodesic in S^n+1. In the second part we study the case of hypersurfaces with constant mean curvature (H ̸= 0). We start by proving a result of comparison between the eigenvalues of the Jacobi operator and the eigenvalue of the Hodge Laplacian, acting in 1-forms, then we use this same technique acting this time in harmonic forms to prove that the Morse index for hypersurfaces with curvature constant mean closed at S^n+1 is bounded inferiorly by a linear function of the first Betti number. We conclude by showing a characterization for the H(r)-torus via the first weak eigenvalue of the Jacobi operator. |
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