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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2604.14537 |
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Sommario:
- 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.