Distorted plane waves on manifolds of nonpositive curvature

TL;DR

This paper analyzes the high frequency behavior of distorted plane waves on nonpositively curved manifolds, providing precise asymptotics, uniform bounds, and insights into nodal set measures, extending previous results to more general geometries.

Contribution

It offers a new high frequency asymptotic expression for distorted plane waves on nonpositively curved manifolds with negative curvature near trapped sets, generalizing prior convex co-compact results.

Findings

Derived a precise high frequency expression for distorted plane waves.

Established uniform $L_{loc}^ abla$ bounds on these waves.

Showed that the nodal sets satisfy a Hausdorff measure estimate similar to Yau's conjecture.

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.