A Spectral Splitting Theorem for the $N$-Bakry Émery Ricci tensor
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908332917260288 |
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| author | Yeung, Wai-Ho |
| author_facet | Yeung, Wai-Ho |
| contents | 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 $γ<\left(\frac{1}{(n-1)\left(1 + \frac{n-1}{N}\right)} + \frac{n-1}{4}\right)^{-1}$ such that $$λ_1(-γΔ_f+\operatorname{Ric}^N_f)\ge 0$$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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Spectral Splitting Theorem for the $N$-Bakry Émery Ricci tensor Yeung, Wai-Ho Differential Geometry Analysis of PDEs 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 $γ<\left(\frac{1}{(n-1)\left(1 + \frac{n-1}{N}\right)} + \frac{n-1}{4}\right)^{-1}$ such that $$λ_1(-γΔ_f+\operatorname{Ric}^N_f)\ge 0$$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. |
| title | A Spectral Splitting Theorem for the $N$-Bakry Émery Ricci tensor |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2504.14962 |