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 $x \in M$?
1
-
$\begingroup$ Bishop-Gromov volume comparison gives a bound in terms of $d(x,x_0)$ and $\nu$, see en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality $\endgroup$– Igor BelegradekCommented Jun 16, 2014 at 15:34
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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.
