A sharp spectral splitting theorem

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 $n\ge 2$ has at least two ends and $${\lambda }_1(-\gamma \mathrm{\Delta }+\mathrm{R}\mathrm{i}\mathrm{c})\ge 0,$$ 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 $\gamma >0$.