(joint work with Semyon Dyatlov and Long Jin)

The eigenmodes of the Laplacian on a smooth compact Riemannian manifold $(M,g)$ can exhibit various localization properties in the high frequency limit, which depend on the properties of the geodesic flow on $(M,g)$. 

I will focus on a "quantum chaotic" situation, namely assume that the geodesic flow is strongly chaotic (Anosov); this is the case if the sectional curvature of $(M,g)$ is strictly negative. The Quantum Ergodicity theorem then states that almost all eigenmodes become equidistributed on $M$ in the the high-frequency limit. The Quantum Unique Ergodicity conjecture states that this behaviour suffers no exception, namely all eigenstates equidistribute in this limit.

This conjecture remaining open, a less ambitious goal is to constrain the possible localization behaviours of the eigenmodes. I will report on recent progress in the case of negative curvature surfaces. Generalizing a previous work by Dyatlov-Jin in the constant curvature case, we show that the eigenmodes cannot concentrate on a proper subset of $M$, in the high frequency limit.

More precisely, any semiclassical measure associated with the sequence of eigenmodes must have full support on $S^*M$. The proof uses the foliation of the phase space into stable and unstable manifolds, methods from semiclassical analysis, and a new Fractal Uncertainty Principle due to Bourgain-Dyatlov, which I will use as a "black box". I plan to describe the strategy of proof in the constant curvature case, and indicate the necessary modifications in the variable curvature setting.