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Semiclassical Measures on Hyperbolic Manifolds

Elena Kim, Nicholas Miller

Abstract

We examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic $(n+1)$-manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic submanifold. Our proof adapts the argument of Dyatlov and Jin to higher dimensions and classifies the closures of horocyclic orbits using Ratner theory. An important step in the proof is a generalization of the higher-dimensional fractal uncertainty principle of Cohen to Fourier integral operators, which may be of independent interest.

Semiclassical Measures on Hyperbolic Manifolds

Abstract

We examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic -manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic submanifold. Our proof adapts the argument of Dyatlov and Jin to higher dimensions and classifies the closures of horocyclic orbits using Ratner theory. An important step in the proof is a generalization of the higher-dimensional fractal uncertainty principle of Cohen to Fourier integral operators, which may be of independent interest.

Paper Structure

This paper contains 35 sections, 54 theorems, 299 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Previous results
  4. Proof outline for Theorem \ref{['thm:support']}
  5. Structure of the paper
  6. Preliminaries
  7. Hyperbolic manifolds
  8. $U_1^-$-dense sets
  9. Semiclassical definitions
  10. Propagating operators
  11. Symbol class
  12. Fourier integral operators
  13. Quantization
  14. Totally geodesic submanifolds and orbit closures
  15. Generalities on geodesic submanifolds and their frame bundles
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $M$ be a compact hyperbolic manifold. If $\mu$ is a semiclassical measure on $M$, then for some $q \in F^*M$,

Figures (1)

  • Figure 1: The left image represents the set from Definition \ref{['def:hyperbolic_ball_porosity']}: $\{e^{\mathcal{V}^{-} v} e^{\mathcal{U}^+ u} q_0: u \in \mathbb{R}^n, |u| \leq \nu \alpha, v \in \mathbb{R}^{n+1}, |v| \leq \varepsilon \}$, where $q_0 = e^{\mathcal{U}^+ u_0} q$. The right image represents the set from Definition \ref{['def:hyperbolic_line_porosity']}: $\{e^{\mathcal{V}^{-} v} e^{\mathcal{U}^+ u} q_1: u \in \mathbb{R}^n, |u| \leq \nu \alpha, v \in \mathbb{R}^{n+1}, |v| \leq \varepsilon \}$, where $q_1 = e^{U^+_1 t_0} q$. The picture is to give intuition only; as $U^+_i$, $U_i^-$, $X$ do not commute, they do not give a coordinate system.

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 92 more