Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alix Deleporte, Jean Lagacé, Marc Rouveyrol

TL;DR

The paper proves a metric-uniform spectral inequality on manifolds with bounded sectional curvature, showing that for any $(M,g)$ in $\\mathcal{M}(d,\\kappa)$ and any thick set $\\omega$, the spectral projection satisfies $\\|\\Pi_\\Lambda u\\|_{L^2(M)} \\le e^{C_{spec}(\\Lambda+1)} \\|u\\|_{L^2(\\omega)}$ with a constant depending only on $d$, $\\kappa$, and thickness parameters, not on the injectivity radius. The authors blend a harmonic-extension method with Logunov–Malinnikova propagation of smallness, working in harmonic coordinates on normal covers to obtain uniform local estimates that piece together into a global inequality; curvature bounds supply the essential volume-doubling and coordinate-control properties. As applications, they derive uniform observability and null-controllability costs for the heat equation across the entire class $\\mathcal{M}(d,\\kappa)$, including fixed-time observability, and they situate their results among prior non-compact and curvature-bounded analyses while extending thickness-based sufficiency to this broad geometric setting. The methods yield a robust geometric-analytic framework based on normal-cover charts and propagation of smallness, enabling a uniform theory insensitive to degenerating injectivity radii. This advances spectral-inequality techniques in curved, non-compact geometries and provides explicit control of constants via dimension, curvature, and thickness.

Abstract

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 $ω$ 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.

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

TL;DR

The paper proves a metric-uniform spectral inequality on manifolds with bounded sectional curvature, showing that for any in and any thick set , the spectral projection satisfies with a constant depending only on , , and thickness parameters, not on the injectivity radius. The authors blend a harmonic-extension method with Logunov–Malinnikova propagation of smallness, working in harmonic coordinates on normal covers to obtain uniform local estimates that piece together into a global inequality; curvature bounds supply the essential volume-doubling and coordinate-control properties. As applications, they derive uniform observability and null-controllability costs for the heat equation across the entire class , including fixed-time observability, and they situate their results among prior non-compact and curvature-bounded analyses while extending thickness-based sufficiency to this broad geometric setting. The methods yield a robust geometric-analytic framework based on normal-cover charts and propagation of smallness, enabling a uniform theory insensitive to degenerating injectivity radii. This advances spectral-inequality techniques in curved, non-compact geometries and provides explicit control of constants via dimension, curvature, and thickness.

Abstract

Given a Riemannian manifold endowed with a smooth metric satisfying upper and lower sectional curvature bounds, we show an equivalence property between the norm on and the norm on subsets 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.
Paper Structure (11 sections, 14 theorems, 76 equations)

This paper contains 11 sections, 14 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Metric-uniform spectral inequality
  3. Applications to observability and control theory
  4. Further remarks and comparison to existing results
  5. Summary of the proof
  6. Geometric preliminaries
  7. Geometric quantities and notation
  8. Volume and density bounds
  9. Normal covers and good coordinate systems
  10. Elliptic tools
  11. Proof of the main theorems

Key Result

Theorem 1.3

Let $d \geq 2$, $\kappa \geqslant 0$ and $R, \varepsilon > 0$. There exists a constant $C_{\rm spec}$ such that for any $(R,\varepsilon)$-thick subset $\omega$ of a Riemannian manifold $(M,g) \in \mathcal{M}(d,\kappa)$ supports a spectral inequality with constant $C_{\rm spec}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.5
  • Proposition 1.6: deleporterouveyrol2024spectralhypsurfaces, Theorem 2
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 23 more