Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle

Abstract

This expository article, written for the proceedings of the Journées EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain and Long Jin. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in L² norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.