Distorted plane waves on manifolds of nonpositive curvature

Abstract

We will consider the high frequency behaviour of distorted plane waves on manifolds of nonpositive curvature which are Euclidean or hyperbolic near infinity, under the assumption that the curvature is negative close to the trapped set of the geodesic flow and that the topological pressure associated to half the unstable Jacobian is negative. We obtain a precise expression for distorted plane waves in the high frequency limit, similar to the one in \cite{GN} in the case of convex co-compact manifolds. In particular, we will show $L_{loc}^\infty$ bounds on distorted plane waves that are uniform with frequency. We will also show that the real part of distorted plane waves restricted to a compact set satisfy the analogue of Yau's conjecture about the Haussdorff measure of nodal sets.