Research Group Geometric structures and moduli spaces

Research on moduli spaces of geometric structures

We present here some research projects funding parts of our research activity.

Some of our projects

We present here the research projects that partially fund some of the group members.

  • Start date

    01/09/2021

  • Duration in months

    36

  • Funding

    FNR

  • Project Team

    Tom Cremaschi, Christian El Emam, Wayne Lam, Nathaniel Sagman, Jean-Marc Schlenker

  • Abstract

    Geometric properties of convex surfaces in Euclidean space are a classical topic in mathematics since the work of Legendre and Cauchy on the rigidity of polyhedra. One of the key results is an isometric embedding theorem (due to Weyl, Lewy, Alexandrov, Pogorelov, etc): any metric of curvature K ≥ 0 on the sphere is induced on the boundary of a unique convex body in R3. This result was extended to surfaces in hyperbolic space by Alexandrov in the 1950s, and a “dual” statement, describing convex bodies in terms of the third fundamental form of their boundary (e.g. their dihedral angles, for an ideal polyhedron) was later discovered. The project’s main focus is to extend the Alexandrov theorem on isometric embeddings of convex surface in H3 and its dual to unbounded convex subsets and convex surfaces, in ways that are relevant to contemporary geometry. One focus is on convex domain having a “thin” asymptotic boundary, for instance a quasicircle – this part of the problem is strongly related to the theory of Kleinian groups. A second direction is convex subsets with a “thick” ideal boundary, for instance a disjoint union of disks – here one find connections to problems in complex analysis, such as the Koebe circle domain conjecture. A third direction is on convex disks of infinite area in H3 and surfaces in hyperbolic ends – with connections to questions on circle packings. Finally the last direction of the project is on analogs of the other topics in anti-de Sitter geometry, a Lorentzian cousin of hyperbolic geometry where interesting new phenomena can occur.

  • Start date

    01/09/2023

  • Duration in months

    24

  • Funding

    Horizon Europe

  • Project Team

    Tom Cremaschi

  • Abstract

    MSCA project on hyperbolic geometry of 3-dimensional manifold, funding the research activity of Tom Cremaschi.