# Control of eigenfunctions on surfaces of variable curvature

## TL;DR

This paper establishes lower bounds on eigenfunction mass, controllability of the Schrödinger equation, and exponential decay of damped wave solutions on variable curvature surfaces with Anosov flows, extending previous constant curvature results.

## Contribution

It extends microlocal eigenfunction bounds and control results from constant to variable negative curvature surfaces with non-smooth foliations.

## Key findings

- Eigenfunctions have full support on such surfaces.
- Controllability holds for the Schrödinger equation with any nonempty open set.
- Exponential energy decay for damped wave equations with nontrivial damping.

## Abstract

We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schr\"odinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works [arXiv:1705.05019], [arXiv:1712.02692], which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of [arXiv:1705.05019], [arXiv:1712.02692] and rely on the fractal uncertainty principle of [arXiv:1612.09040]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used in [arXiv:1504.06589]. Instead, our argument uses Egorov's Theorem up to local Ehrenfest time and the hyperbolic parametrix of [arXiv:0706.3242], together with the $C^{1+}$ regularity of the stable/unstable foliations.