Spectral estimates on hyperbolic surfaces and a necessary condition for observability of the heat semigroup on manifolds

TL;DR

This paper extends spectral estimate results to hyperbolic surfaces with cusps, establishing a necessary thickness condition for observability of the heat semigroup, using propagation of smallness techniques.

Contribution

It generalizes spectral estimates to hyperbolic surfaces with cusps and proves the necessity of a thickness condition for observability.

Findings

Spectral estimates hold under a thickness condition on sets.

The thickness condition is necessary for observability on manifolds with Ricci curvature bounds.

Propagation of smallness estimates are used in the proofs.

Abstract

This article is a continuation of arXiv:2401.14977. We study the concentration properties of spectral projectors on manifolds, in connection with the uncertainty principle. In arXiv:2401.14977, the second author proved an optimal uncertainty principle for the spectral projector of the Laplacian on the hyperbolic half-plane. The aim of the present work is to generalize this condition to surfaces with hyperbolic ends. In particular, we tackle the case of cusps, in which the volume of balls of fixed radius is not bounded from below. We establish that spectral estimates hold from sets satisfying a thickness condition, with a proof based on propagation of smallness estimates of Carleman and Logunov--Malinnikova type. We also prove the converse, namely the necessary character of the thickness condition, on any smooth manifold with Ricci curvature bounded from below.