In the paper 


*The volume of a small geodesic ball in a Riemannian manifold*, Michigan Mathematical Journal **20** (1973), doi: [10.1307/mmj/1029001150](http://dx.doi.org/10.1307/mmj/1029001150)


A. Gray proved the following result:



> 
> Let $M$ be a Riemannian manifold of dimension $n$ with positive scalar Ricci curvature at $x \in M$. If $R$ is a sufficiently small real number, calling $B(x, \, R)$ a geodesic ball of radius $R$ centered at $x$, we have $$\mathrm{vol} \,(B(x, \, R)) \leq C R^n,$$ where $C$ is a constant depending only on $n$.
> 
> 
> 


My question involves balls of sufficiently big radius, instead of sufficiently small one. In other words, let me as what follows:



> 
> Let $M \subset \mathbb{R}^N$ be an (unbounded) embedded submanifold of dimension $n$. Under which conditions on $M$ and $x \in M$ it is possible to find constants $C$ and $\gamma$ such that $$\mathrm{vol} \,(B(x, \, R)) \leq C R^{\gamma}$$ 
>  for all $R >>0$? 
> 
> 
> 


I am not an expert on differential geometry, so I apologize in advance if my formulation of the problem is not optimal, or if the answer turns out to be somehow trivial.


Any reference to the relevant literature will be greatly appreciated.



Volume of large balls in embedded submanifolds
Since volume is an intrinsic geometric invariant, the fact that $M$ is a submanifold is irrelevant. If you assume that Ricci curvature is nonnegative, then the inequality follows from the [Bishop-Gromov inequality](http://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality) (in fact, just the Bishop inequality).


Also, a more general condition is given in another [MathOverflow question](https://mathoverflow.net/questions/283748/bishop-gromov-volume-comparison-on-manifolds-with-negligible-negative-ricci-curv). Even this condition can probably be weakened more, but there has to be some kind of assumption saying "not too much negative Ricci curvature".


