Doctoral Thesis
DOI
10.11606/T.45.2019.tde-15022019-111803
Document
Author
Full name
Carlos Wilson Rodríguez Cárdenas
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
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
Bifurcation
Bumpy metrics
Constant mean curvature
Free boundary
Jacobi operator
Stability
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
Bifurcação.
Curvatura Média Constante
Fronteira Livre
Métricas Bumpy