A Spectral Splitting Theorem for the $N$-Bakry \'Emery Ricci tensor

Abstract

We extend the spectral generalization of the Cheeger-Gromoll splitting theorem to smooth metric measure space. We show that if a complete non-compact weighted Riemannian manifold $(M,g,e^{-f}\,dvolg)$ of dimension $n\ge 2$ has at least two ends where $f$ is smooth and bounded. If there is some $N\in (0,\infty)$ and $\gamma<\left(\frac{1}{(n-1)\left(1 + \frac{n-1}{N}\right)} + \frac{n-1}{4}\right)^{-1}$ such that $$\lambda_1(-\gamma \Delta_f+\operatorname{Ric}^N_f)\ge 0$$then $M$ splits isometrically as $\mathbb{R}\times X$ for some complete Riemannian manifold $X$ with $(\operatorname{Ric}_X)^N_f\ge 0$. The estimate can recover the spectral splitting result and its sharp constant $\frac{4}{n-1}$ in Antonelli-Pozzetta-Xu and and Catino--Mari--Mastrolia--Roncoroni.