Control of eigenfunctions on surfaces of variable curvature

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.