Seminar

When $M$ is a hyperbolic $3$-manifold with $\partial M \neq \varnothing$, the geometric structure on $M$ can be deformed by varying the conformal structure on $\partial M$. If we further assume that the injectivity radius in the convex core of $M$ is bounded both above and below, then any such deformation decays exponentially fast inside the convex core. This phenomenon, known as \emph{geometric inflexibility}, is especially striking when $M$ is geometrically infinite: as one goes deeper into the degenerate end, the deformation becomes exponentially close to an isometry.

A beautiful application of this inflexibility, due to Brock and Bromberg, is the proof of convergence of the iterates of a pseudo-Anosov map on quasi-Fuchsian space. This, in turn, implies the hyperbolization theorem for $3$-manifolds that fiber over the circle with pseudo-Anosov monodromy. In this talk, I will explain the idea of geometric inflexibility and sketch the proof of this convergence result.