Uma caracterização espectral para os H(r)-toros na esfera

• Ano de defesa: 2020
• Autor(a) principal: Carlos Alberto Cjanahuiri Aroquipa
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Minas Gerais; Brasil; ICX - DEPARTAMENTO DE MATEMÁTICA; Programa de Pós-Graduação em Matemática; UFMG
• Palavras-chave em Português: 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
• Link de acesso: http://hdl.handle.net/1843/44583

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.