Preprint
Inserted: 18 dec 2024
Pages: 10
Year: 2024
Abstract:
We prove a sharp spectral generalization of the Cheeger--Gromoll splitting theorem. We show that if a complete non-compact Riemannian manifold M of dimension $n\geq 2$ has at least two ends and $\lambda_1(-\gamma\Delta+\mathrm{Ric})\geq 0$ for some $\gamma<\frac{4}{n-1}$, then M splits isometrically as $\mathbb R\times N$ for some compact manifold $N$ with nonnegative Ricci curvature. We show that the constant $\frac{4}{n-1}$ is sharp, and the multiple-end assumption is necessary for any $\gamma>0$.