Convex Surfaces in Hyperbolic Geometry (CoSH)

We intend to extend classical results on convex surfaces in the hyperbolic space to convex surfaces are infinite.

The project at a glance

Start date:

01 Sep 2021
Duration in months:

36
Funding:

FNR OPEN
Principal Investigator(s):

Jean-Marc Schlenker

About

We intend to extend classical results on convex surfaces in the hyperbolic space to convex surfaces are infinite. The project is centered on four wide conjectures which are shown to be connected to important problems in diverse areas of contemporary mathematics and mathematical physics. 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 H 3 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 H 3 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.