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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2604.14537 |
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| _version_ | 1866910133357903872 |
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| author | Navarro, Dimitri Pan, Jiayin Zhu, Xingyu |
| author_facet | Navarro, Dimitri Pan, Jiayin Zhu, Xingyu |
| contents | For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically
$$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$
Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s^2$ as $s\to\infty$, then $π_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+δ}$ for some $δ\in (0,1)$ and the Ricci curvature is positive at a point, then $π_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14537 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete manifolds with nonnegative Ricci curvature and slow relative volume growth Navarro, Dimitri Pan, Jiayin Zhu, Xingyu Differential Geometry For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s^2$ as $s\to\infty$, then $π_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+δ}$ for some $δ\in (0,1)$ and the Ricci curvature is positive at a point, then $π_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth. |
| title | Complete manifolds with nonnegative Ricci curvature and slow relative volume growth |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.14537 |