Spectral geometry of embeddings and fibre bundles

• Grant number: 23/04597-3
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2023
• End date: October 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Paolo Piccione
• 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

Abstract

We intend to extend the theory of critical metrics for the first eigenvalue of the Laplacian developed by El Soufi and Ilias in two directions: to consider this functional restricted to the class of metrics induced by embeddings into a fixed ambient manifold, and also to consider the other eigenvalues in this setting. Moreover, we intend to study the spectral geometry of surface bundles over the circle and what are the necessary and sufficient conditions to prescribe the first eigenvalue of the Laplacian of the fibres. Finally, we intend to apply bifurcation techniques to investigate what occurs to the first eigenvalue of the Laplacian in the conformal classes of Berger metrics.