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.
Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum / L. Mari, M. Ranieri, E. Sampaio, F. Vitório. - (2025 Jun).
Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum
L. MariPrimo
;
2025
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.| File | Dimensione | Formato | |
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