Convex hulls of limits sets of hyperbolic and AdS manifolds

Goal: The ultimate goal of the project is to visualize limit sets of "quasifuchsian hyperbolic manifolds" and "globally hyperbolic AdS manifolds". Each of those objects is determined by a homeomorphism from a discrete group G, defined by 2g generators and one relation only, to the group SO(3,1), respectively SO(2,2), of 4x4 matrices preserving a symmetric bilinear form of signature (3,1), respectively (2,2). Examples of such homeomorphisms can be computed in a number of ways.

Once those homeomorphism are computed, one should be able to compute and visualize their "limit sets", which are Jordan curves (typically non-regular, and fractal) either in the sphere or in a one-sheeted hyperboloid in R3. One way to visualize those limit sets is by computing their convex hull, which have remarkable geometric properties.

Visualization could be achieved either by producing "movies" of how limit sets vary when one varies some parameters of the homeomorphims defining them, or by printing 3d models of their convex hulls for different values of some parameters. It should be interesting to compare the results in the "hyperbolic" and the "AdS" situations.

Schedule: to be determined.

Persons involved: Advised by Jean-Marc Schlenker, NN.

Difficulty level: medium/advanced.

Tools: programs in python/sage, use of computational tools to compute the first eigenvalues of the Laplace operator.

Expected results: either disprove the conjecture, or give some evidence that it might be correct by showing that, up to some approximation, the first few eigenvalues of the Laplace operator remain constant in an isometric deformation in a few simple examples.

FSCT -- University of Luxembourg