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Doctoral Thesis
Full name
Carlos Wilson Rodríguez Cárdenas
Knowledge Area
Date of Defense
São Paulo, 2018
Piccione, Paolo (President)
Earp, Henrique Nogueira de Sá
Manfio, Fernando
Mossa, Roberto
Siciliano, Gaetano
Title in English
Genericity of bumpy metrics, bifurcation and stability in free boundary CMC hypersurfaces
Keywords in English
Bumpy metrics
Constant mean curvature
Free boundary
Jacobi operator
Abstract in English
In this thesis we prove the genericity of the set of metrics on a manifold with boundary M^{n+1}, such that all free boundary constant mean curvature (CMC) embeddings \varphi: \Sigma^n \to M^{n+1}, being \Sigma a manifold with boundary, are non-degenerate (Bumpy Metrics), (Theorem 2.4.1). We also give sufficient conditions to obtain a free boundary CMC deformation of a CMC inmersion (Theorems 3.2.1 and 3.2.2), and a stability criterion for this type of immersions (Theorem 3.3.3 and Corollary 3.3.5). In addition, given a one-parametric family, {\varphi _t : \Sigma \to M} , of free boundary CMC immersions, we give criteria for the existence of smooth bifurcated branches of free boundary CMC immersions for the family {\varphi_t}, via the implicit function theorem when the kernel of the Jacobi operator J is non-trivial, (Theorems 4.2.3 and 4.3.2), and we study stability and instability problems for hypersurfaces in this bifurcated branches (Theorems 5.3.1 and 5.3.3).
Title in Portuguese
Genericidade das métricas bumpy, bifurcação e estabilidade em hipersuperfícies de CMC e fronteira livre
Keywords in Portuguese
Curvatura Média Constante
Fronteira Livre
Métricas Bumpy
Operador de Jacobi
Abstract in Portuguese
Nesta tese, provamos a genericidade do conjunto de métricas em uma variedade com fronteira M^{n+1}, de modo que todos os mergulhos de curvatura média constante (CMC) e fronteira livre \varphi : \Sigma^n \to M^{n+1}, sendo \Sigma uma variedade com fronteira, sejam não-degenerados (Métricas Bumpy), (Teorema 2.4.1). Nós também fornecemos condições suficientes para obter uma deformação CMC e fronteira livre de uma imersão CMC (Teoremas 3.2.1 and 3.2.2), e um critério de estabilidade para este tipo de imersões (Teorema 3.3.3 and Corolario 3.3.5). Além disso, dada uma família 1-paramétrica, {\varphi _t : \Sigma \to M} , de imersões de CMC e fronteira livre, damos os critérios para a existência de ramos de bifurcação suaves de imersões CMC e fronteira livre para a familia {\varphi_t}, por meio de o teorema da função implícita quando o kernel do operador Jacobi J é não-trivial, (Teoremas 4.2.3 and 4.3.2), e estudamos o problema da estabilidade e instabilidade para hipersuperfícies em naqueles ramos de bifurcação (Teoremas 5.3.1 and 5.3.3).
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