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Let $(M,g)$ be a complete Riemannian manifold and suppose $\mathrm{Ric}(g) \geq -k$ for some $k>0$. Suppose we know that $\mathrm{vol}_g (B_1^g (x_0)) \geq \nu$ for some particular $x_0 \in M$ and some $\nu >0$. Does this information tell us anything about a lower bound on $\mathrm{vol}_g (B_1^g (x))$ for other $x \in M$?

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No, for cusped hyperbolic manifolds you can have arbirtrarily small volumes of unitary balls.

In general also if $M$ is compact you can have very large balls and very thin margulis tubes.

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