A sharp spectral splitting theorem

TL;DR

This paper proves a precise spectral splitting theorem for non-compact Riemannian manifolds with multiple ends, establishing conditions under which the manifold splits isometrically, and demonstrates the sharpness of the spectral constant involved.

Contribution

It provides a sharp spectral generalization of the Cheeger--Gromoll splitting theorem, identifying exact conditions for manifold splitting based on spectral bounds and end structure.

Findings

Manifolds with at least two ends and spectral condition split as R x N.

The constant 4/(n-1) is proven to be optimal.

Multiple-end assumption is necessary for positive gamma.

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 $\gamma>0$.