Control of eigenfunctions on surfaces of variable curvature

We prove a microlocal lower bound on the mass of high energy eigenfunctions Laplacian compact surfaces negative curvature, and more generally with Anosov geodesic flows. This implies controllability for Schrödinger equation by any nonempty open set, shows that every semiclassical measure has full support. also exponential decay solutions to damped wave such surfaces, nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov Long Jin [Acta Math. 220 (2018), pp. 297–339] [Comm. Phys. 373 (2020), 771–794]), which considered setting constant curvature. The proofs use strategy 297–339 771–794] rely fractal uncertainty principle Jean Bourgain [Ann. (2) 187 825–867]. However, in variable curvature case stable/unstable foliations are not smooth, so we can no longer associate these pseudodifferential calculus type used Joshua Zahl [Geom. Funct. Anal. 26 (2016), 1011–1094]. Instead, our argument uses Egorov’s theorem up local Ehrenfest time hyperbolic parametrix Stéphane Nonnenmacher Maciej Zworski 203 (2009), 149–233], together C 1 + C^{1+} regularity foliations.