Genericity of bumpy metrics, bifurcation and stability in free boundary CMC hypersurfaces

Author(s)

Carlos Wilson Rodríguez Cárdenas

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2018-12-03

Examining board members

Paolo Piccione; Henrique Nogueira de Sá Earp; Fernando Manfio; Roberto Mossa; Gaetano Siciliano

• Advisor: Paolo Piccione

Abstract

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). (AU)

• FAPESP's process: 17/10001-5 - Topics in bifurcation theory for geometric variational problems
• Grantee: Carlos Wilson Rodriguez Cardenas
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)