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

TL;DR

This paper establishes a spectral inequality on Riemannian manifolds with bounded sectional curvature, linking global and localized norms of spectral functions, independent of the injectivity radius.

Contribution

It proves a metric-uniform spectral inequality for the Laplacian on manifolds with curvature bounds, highlighting independence from the injectivity radius.

Findings

Norm equivalence depends only on curvature bounds, dimension, and spectral cutoff.

The thickness condition on subsets is proven to be optimal.

The inequality holds uniformly across manifolds with bounded sectional curvature.

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 $\omega$ 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.