# Covering hyperbolic manifolds by round balls

This question is a natural follow up to the question asked here. I think it should not be too hard to answer it (negatively) in dimension 3, but higher dimensions will be probably challenging.

Let $$M$$ be a connected hyperbolic $$n$$-manifold. A round ball in $$M$$ is an open metric $$R$$-ball $$B=B(p,R)\subset M$$ which admits a Riemannian isometry to an open $$R$$-ball in the hyperbolic space $${\mathbb H}^n$$. In other words, the exponential map $$\exp_p: T_pM\to M$$ is a diffeomorphism from $$B(0,R)\subset T_pM$$ to $$B$$.

Question. Is there a function $$V(n,m)$$ such that for every compact hyperbolic $$n$$-manifold $$M$$, $$n\ge 3$$, covered by $$m$$ round metric balls, the volume of $$M$$ is $$\le V(n,m)$$?

Note that the answer is negative in the case of hyperbolic surfaces (even $$m=3$$ suffices), see here.