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

TL;DR

This paper demonstrates that eigenfunctions of the Laplacian on hyperbolic surfaces have a uniform lower bound in their $L^2$ norm on any nonempty open set, using the fractal uncertainty principle.

Contribution

It applies the fractal uncertainty principle to establish uniform lower bounds for Laplacian eigenfunctions on hyperbolic surfaces, advancing understanding of their localization properties.

Findings

Eigenfunctions are bounded below in $L^2$ norm on open sets.

The bounds are independent of eigenvalues.

Application of fractal uncertainty principle to hyperbolic geometry.

Abstract

This expository article, written for the proceedings of the Journ\'ees EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [arXiv:1612.09040] and Long Jin [arXiv:1705.05019]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in $L^2$ norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.