A sharp spectral splitting theorem

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 $n\ge 2$ has at least two ends and \[ λ_1(-γΔ+\mathrm{Ric})\geq 0, \] for some ${\hat{I}}^3<\frac{4}{n-1}$, 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 ${\hat{I}}^3>0$.

10 pages. Comments welcome!