arXiv subjects

Marc Rouveyrol

Publications and source records attributed to Marc Rouveyrol.

4 recordsLinked to original sources

arXiv2026

Metric-uniform spectral inequality for the Laplacian on manifolds with bounded sectional curvature

Given a Riemannian manifold $M$ endowed with a smooth metric $g$ satisfying upper and lower sectional curvature bounds, we show an equivalence property between the $\mathrm{L}^2$ norm on $M$ and the $\mathrm{L}^2$ norm on subsets $\omega$ satisfying a thickness condition, for functions in the range of a spectral projector. The thickness condition is known to be optimal in this setting. The constant appearing in the equivalence of norms property depends only on the dimension of the manifold, curvature bounds, and frequency threshold of the spectral cutoff, but, crucially, not on the injectivity radius.

math.AP↗

arXiv2024

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

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.

math.AP↗

arXiv2024

Stabilization of the wave equation on larger-dimension tori with rough dampings

This paper deals with uniform stabilization of the damped wave equation. When the manifold is compact and the damping is continuous, the geometric control condition is known to be necessary and sufficient. In the case where the damping is a sum of characteristic functions of polygons on a two-dimensional torus, a result by Burq-Gérard states that stabilization occurs if and only if every geodesic intersects the interior of the damped region or razes damped polygons on both sides. We give a natural generalization of their result to a sufficient condition on tori of any dimension $d \geq 3$. In some particular cases, we show that this sufficient condition can be weakened.

math.AP↗

arXiv2024

Spectral estimate for the Laplace-Beltrami operator on the hyperbolic half-plane

The purpose of this note is to investigate the concentration properties of spectral projectors on manifolds. This question has been intensively studied (by Logvinenko--Sereda, Nazarov, Jerison--Lebeau, Kovrizhkin, Egidi--Seelmann--Veseli{\'c}, Burq--Moyano, among others) in connection with the uncertainty principle. We provide the first high-frequency results in a geometric setting which is neither Euclidean nor a perturbation of Euclidean. Namely, we prove the natural (and optimal) uncertainty principle for the spectral projector on the hyperbolic half-plane.

math.AP↗