A sharp spectral splitting theorem
Fuente:
arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912159790792704 |
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| author | Antonelli, Gioacchino Pozzetta, Marco Xu, Kai |
| author_facet | Antonelli, Gioacchino Pozzetta, Marco Xu, Kai |
| contents | 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
\[
λ_1(-γΔ+\mathrm{Ric})\geq 0,
\]
for some $γ<\frac{4}{n-1}$, 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 $γ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A sharp spectral splitting theorem Antonelli, Gioacchino Pozzetta, Marco Xu, Kai Differential Geometry Analysis of PDEs 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 \[ λ_1(-γΔ+\mathrm{Ric})\geq 0, \] for some $γ<\frac{4}{n-1}$, 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 $γ>0$. |
| title | A sharp spectral splitting theorem |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2412.12707 |