It is well known that a lower bound on Ricci curvature gives an upper bound on the volume of balls. What are conditions that gives a lower bound on the volume of balls? It is reasonable to think that an upper bound on curvature would give some kind of lower bound on the volume of balls, is this not true? If not, what are some counter-examples?
Here are two results I found in Schoen and Yau's book on Differential Geometry:
1) Given an upper bound on sectional curvature $K_M \leq b^2$, if the metric at $x \in M$ in normal coordinates has the form $ds^2 = dr^2 + r^2g_{ij}(r,\theta)d\theta^id\theta^j$, then $Vol(B_x(R)) \geq C(n,b,R)$ (this constant is independant of $x$). This is on page 11. I don't really understand the condition on the metric, I guess this says that the metric is "comparable to Euclidean metric" somehow? However, they don't say what conditions should hold on those $g_{ij}$'s. Obviously, one could just write $$dr^2 + \tilde{g}_{ij}(r,\theta)d\theta^id\theta^j = dr^2 + r^2(r^{-2}\tilde{g}_{ij}(r,\theta))d\theta^id\theta^j$$ and take $g_{ij} = r^{-2}\tilde{g}_{ij}$ but then the condition on the metric is vacuous...
On page 29, they have another result:
2) Given a lower bound on Ricci, $Ric \geq -(n-1)K$ and a point $p \in M$, we have a lower bound $Vol(B_x(1)) \geq e^{-C\rho(x)}$ for $C = C(n,K,Vol(B_p(1)))$ and $\rho(x) = d(x,p)$. This is interesting but it is far from uniform in $x$.
Do you know of some other results of this kind?