Concentration of symmetric eigenfunctions

TL;DR

This paper investigates how high-frequency eigenfunctions of the Laplace operator concentrate and oscillate on compact Riemannian manifolds, analyzing the structure of their invariant semiclassical measures and their relation to symmetries.

Contribution

It provides new methods to derive invariant semiclassical measures from eigenfunctions with specific symmetries and proves that all geodesic flow-invariant measures are semiclassical measures in constant curvature manifolds.

Findings

Invariant semiclassical measures can be obtained from eigenfunctions with prescribed symmetries.

In manifolds of constant positive curvature, all geodesic flow-invariant measures are semiclassical measures.

Abstract

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.