Is the converse to the [Bishop-Gromov Inequality](http://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality) true? 


In other words, if, for a complete $n$-dimensional Riemannian manifold $M$, there is $k \in \mathbb{R}$, such that defining $V\_k(R)$ to be the size of a ball of radius $R$ in the standard space $S^n\_k$, for any $p \in M$ and $R\geq 0$ we have that 


$$
vol\_M(B\_R(p)) \leq V\_k(R)
$$


then is it true that on $M$, $Ric \geq (n-1)k$?



Converse to Bishop-Gromov Inequality
This is false for $n=3$, and for $k<0$. [One can get a different pinching condition](http://www2.math.uic.edu/~agol/blog/031028.pdf), $R(g)g-Ric \geq -4g$ which is weaker than $Ric \geq -2g$ ($R(g)$ is the scalar curvature), and which gives the same upper bound on volume as Gromov's inequality (for a manifold with negative curvature or no cut points). The idea is that in the proof of Gromov's inequality, one can integrate out the curvature tangent to the sphere of radius $r$ using the theorem egregrium and Gauss-Bonnet, giving a weaker curvature condition. There are metrics which satisfy the first curvature condition but not the Ricci lower bound. For example, there are homogeneous negatively curved metrics on $R^3$ which in one direction are foliated by totally geodesic hyperbolic planes of curvature $-K\_1^2$, and in a perpendicular direction are foliated by totally geodesic hyperbolic planes of curvature $-K\_2^2$ for $0 < K\_1\neq K\_2>0$ (this is a double warped product, and is Thurston's "ninth geometry" of the form $dr^2+e^{2K\_1r}dx^2+e^{2K\_2r}dy^2$). One obtains a counterexample for the choice of $K\_1 =5/4,K\_2=1/2$. 


**Added computations:** The curvature operator of this metric may be normalized to have sectional curvatures $-K\_1^2, -K\_2^2, -K\_1K\_2$. If we assume that $K\_1\geq K\_2$, then the minimal curvature is $-K\_1^2$. Under the same assumption, the minimal eigenvalue of the Ricci operator is $-K\_1^2-K\_1K\_2$. The minimal eigenvalue of $Rg-Ric$ is $-2K\_1^2-K\_1K\_2-K\_2^2$. We then see that for $K\_1=5/4, K\_2=1/2$, the minimal eigenvalue of $Ric$ is $-35/16<-2$, and the minimal eigenvalue of $Rg-Ric$ is $-4$. 


