A Spectral Splitting Theorem for the $N$-Bakry Émery 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 $${\lambda }_1(-\gamma {\Delta }_f+{\mathrm{Ric}}_f^N)\ge 0$$then $M$ splits isometrically as $\mathbb{R}\times X$ for some complete Riemannian manifold $X$ with $({\mathrm{Ric}}_X{)}_f^N\ge 0$. 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.

20 pages. this article draws heavily from arXiv:2412.12707. Change: Correct typos, mistakes in eq(3.28) and latter estimate in [v1]. Adding a remark in the end of section 1