From geometric analysis to bifurcation

• Grant number: 21/03599-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: May 01, 2021
• End date: February 27, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Marcos Martins Alexandrino da Silva
• Grantee: Eduardo Rosinato Longa
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis, AP.TEM
• Associated scholarship(s): 21/09650-4 - Bifurcation of minimal surfaces and the first eigenvalue of the Laplacian, BE.EP.PD

Abstract

We seek to extend the application of the techniques of closed minimal surfaces to the case where they are noncompact and have a boundary, and to explore bifurcation phenomena related to these surfaces in homogeneous 3-manifolds. More specifically, we intend to address the following topics: possible elaboration of a survey about homological systoles in Riemannian manifolds, with the proposal of problems that involve ambients with group actions and generalised geometries, such as Randers metrics (Finsler); to explicitly control the topology of noncompact minimal surfaces with boundary and low index in some 3-dimensional ambients; and to apply bifurcation techniques to prove the abundancy of minimal tori in Berger spheres which are noncongruent to the Clifford torus. (AU)