Deformation spaces of geometric structures – EMERGE

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 crossroad of several disciplines, such as differential and algebraic geometry, low-dimensional topology, representation theory, number theory, real and complex analysis, which makes the subject extremely rich and fascinating.

This project aims to achieve innovative advances towards three main challenges in the field of pseudo-Riemannian geometric structures:

i) the study of quasi-Fuchsian hyperbolic manifolds by means of minimal and constant mean curvature surfaces, in particular leading to the solution of two longstanding conjectures going back to Thurston and Andrews;

ii) the study of spacelike submanifolds of the pseudo-hyperbolic space, achieving existence results and quantitative estimates under certain curvature assumptions; this includes the case of Anti-de Sitter space which has important applications to Teichmüller theory;

iii) the understanding of deformation spaces, by means of the construction of metrics of (para)-hyperKähler type and the investigation of their geometric properties.

I propose a novel approach for the study of geometric structures which, supported by preliminary results, will develop an analytic framework integrating three types of techniques: a priori estimates for submanifolds, geometric flows, and infinite-dimensional reductions.

The proposed methodology and expected results will reveal the huge potential, as yet largely unexplored, of analytic techniques in geometric topology. In the long term, it will find applications in various related areas, namely: the study of higher dimensional pseudo-hyperbolic manifolds, manifolds of variable negative curvature, and the deformation spaces of other types of geometric structures, such as real projective structures.