A spectral inequality for a set $\omega$ tells us quantitatively how small (linear combinations of) eigenfunctions can be on $\omega$. In many settings, it is known that a spectral inequality holding for $\omega$ is equivalent to $\omega$ being thick, which a notion of uniformity.
For manifolds, it was shown by Deleporte and Rouveyrol that a subset $\omega$ of a manifold with Ricci curvature bounded below can only support a spectral inequality if it is thick. Using a quantitative unique continuation result for the gradient of a harmonic function due to Logunov and Malinnikova, we show that on manifolds with bounded sectional curvature, any thick $\omega$ supports a spectral inequality. Crucially, this holds even for manifolds whose injectivity radius goes to zero. I will discuss how this hypothesis came to be removed, and how one may try to match the curvature conditions.
Joint work with A. Deleporte (Paris-Saclay) and M. Rouveyrol (Bielefeld)
