Geometry and analysis for (G,X)-structures and their deformation spaces

Introduction

The study of geometric structures on manifolds finds its inspiration in Klein’s Erlangen Program from 1872, and has seen spectacular developments and applications in geometric topology since the work of Thurston at the end of the 20th century. Geometric structures lie at the crossroads of several disciplines, such as differential and algebraic geometry, low-dimensional topology, representation theory, number theory, and real and complex analysis, which makes the subject extremely rich and fascinating.

Research Context

In the context of geometric structures of pseudo-Riemannian type, the study of submanifolds with special curvature conditions has been very effective and led to some fundamental questions, such as the open conjectures of Andrews and Thurston from the 2000s, and the recently settled Labourie’s Conjecture.

Project Goals

This project aims to obtain important results in this direction, towards four interconnected goals:

1. The study of quasi-Fuchsian hyperbolic manifolds, in particular leading to the proof of a strong statement that would imply the solution of the conjectures of Andrews and Thurston.
2. The achievement of curvature estimates of L²-type on surfaces in Anti-de Sitter space.
3. The construction of metrics of (para)-hyperKähler type on deformation spaces of (G,X)-structures, and the investigation of their properties.
4. The study of existence and uniqueness of special submanifolds of dimension greater than 2 in pseudo-Riemannian symmetric spaces.

Methodology

The project adopts an innovative approach integrating geometric and analytic techniques, and the results will have remarkable applications for Teichmüller theory and Anosov representations.

Long-term Impact

In the long term, the proposed methodology and the expected results will lead to further developments in various related directions, for instance:

• The study of pseudo-Riemannian manifolds of variable negative curvature.
• The study of higher dimensional pseudo-hyperbolic manifolds.
• The deformation spaces of other types of (G,X)-structures.