Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum

TL;DR

This paper characterizes the spectrum of certain non-compact manifolds with Ricci curvature bounds, showing that maximal bottom spectrum implies the manifold's spectrum matches that of hyperbolic space, indicating a form of spectral rigidity.

Contribution

It establishes spectral rigidity results for manifolds with Ricci curvature bounded below and maximal bottom spectrum, extending understanding of geometric and spectral properties.

Findings

Spectrum of manifold matches hyperbolic space when bottom spectrum is maximal.

Spectral rigidity holds for manifolds with Ricci bounds and infinite volume ends.

Abstract

We investigate the spectrum of the Laplacian on complete, non-compact manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1 \in L^1(\infty)$. We prove that if the bottom spectrum attains the maximal value $\frac{(n-1)^2}{4}$ compatible with the curvature bound, then the spectrum of $M$ coincides with that of hyperbolic space $\mathbb{H}^n$, namely, $\sigma(M) = \left[ \frac{(n-1)^2}{4}, \infty \right)$. The result can be localized to an end $E$ with infinite volume.