Geometric analysis and variational problems in Riemannian and Kähler geometry

• Grant number: 15/50470-9
• Support Opportunities: Regular Research Grants
• Start date: June 01, 2016
• End date: December 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Agreement: University of Maryland
• Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
• Principal Investigator: Paolo Piccione
• Grantee: Paolo Piccione
• Principal researcher abroad: Yanir A. Rubenstein
• Institution abroad: University of Maryland, College Park, United States
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 11/21362-2 - Group actions, submanifold theory and global analysis in Riemannian and pseudo-Riemannian geometry, AP.TEM

Abstract

The aim of this Project is to bring together groups of geometers and analysts working at University of Maryland (UMD) and the Universidade de São Paulo (USP) to explain their methods and techniques, survey recent developments and disseminate their own research findings relating broadly to the area of differential geometry, chart new directions of research, and explore possible collaborations. The central theme of the project is built around the Yamabe problem and its generalizations. One research line aims at establishing bifurcation results for solutions of the singular Yamabe problem on spheres that are singular along a sphere of less dimension. More generally, it is suggested to study the existence of multiple solutions of the Yamabe singular problem in manifolds M, that are singular along a codimension 2 (or more) submanifold L whose normal bundle is trivial and conformally equivalent to M/L. A second line of research aims at exploring the analog of edge metrics for the Yamabe problem, about whose existence very little is known. The development of this project requires techniques from Variational Calculus, Kähler geometry and geometric microlocal analysis. (AU)