Séance Séminaire

Séminaire Gaston Darboux

On hyperbolic 3-manifolds with polyhedral boundary

It is known that convex bodies in the model 3-spaces of constant curvature are rigid with respect to the induced intrinsic metric on the boundary. This story has two classical chapters: the rigidity of convex polyhedra and the rigidity of smooth convex bodies, though there is also a common generalization obtained by Pogorelov. Similarly to this, Jean-Marc Schlenker proved that hyperbolic metrics with smooth strictly convex boundary on a compact hyperbolizable 3-manifold M are rigid with respect to the induced metric on the boundary (and also with respect to the dual metric). It is reasonable to expect that similar results should hold also for polyhedral boundaries, and eventually for general convex boundaries. Curiously enough, no polyhedral counterparts were proven up to now, though some partial progress was made by François Fillastre. One of the main difficulties is that convex hyperbolic cone-metrics on the boundary of M (which is a standard intrinsic description of what we expect to be the induced metric on a polyhedral boundary) might admit not so polyhedral realizations, which are hard to handle or to exclude. A prototypical example is the boundary of a convex core bent along an irrational lamination. I will present a recent work proving the rigidity (and the dual rigidity) of hyperbolic metrics on M with convex polyhedral boundary under mild additional assumptions. As another outcome, it follows that convex cocompact hyperbolic metrics on the interior on M with the convex cores that are "almost polyhedral" are globally rigid with respect to the induced metric on the boundary of the convex core, and are infinitesimally rigid with respect to the bending lamination. This is a step towards conjectures of William Thurston.