GAPR

Inspired by the famous Erlangen Program from 1872, where Felix Klein promoted the idea that geometries are governed by their group of symmetries, the study of (G,X)-structures on manifolds has been developed in the second half of the 20th century by, among others, Charles Ehresmann and William Thurston. Several types of Riemannian geometric structures played a fundamental role in the proof of the Geometrization Conjecture from 2003, and among those the richest and most intriguing ones are the hyperbolic structures. Recently, a large interest has grown for various generalizations to non-Riemannian settings, such as pseudo-hyperbolic structures, real projective structures, and many others. The goal of this project is to develop analytic methods in geometric topology, and apply them to the study of hyperbolic and pseudo-hyperbolic structures. Two main directions will be explored, each corresponding to an analytic method, namely geometric flows and a priori curvature estimates. For the former, we expect applications of geometric flows in a pseudo-Riemannian setting, such as the mean curvature flow, to have a great potential in the context of geometric structures, such as, but not limited to, providing a proof of a conjecture of Ben Andrews on almost-Fuchsian three-dimensional hyperbolic manifolds. For the latter, we aim at proving results that relate the geometric properties (such as the curvature) of special submanifolds (for instance maximal and constant mean curvature surfaces) to their asymptotic behaviour in pseudo-hyperbolic spaces; these estimates will have remarkable applications in geometric function theory and Teichmüller theory. Besides the novel results that are expected, the project aims at demonstrating the potential, still largely unexplored, of analytic methods for (G,X)-structures, thus having a wide long-term impact both on the geometric topology and geometric analysis communities.