Suppose $(M,g)$ is a simply connected, complete Riemannian $N$-manifold with negative curvature, with its Ricci curvature bounded from below and for all $x\in M$ we have $\text{Vol}_g(B_1(x))\geq v$ for some $v>0$. ($B_1(x)$ being the unit geodesic ball with center at $x\in M$)
My question is, whether such a manifold could exist. And if it exists, can we construct such a manifold suitably? Any help is appreciated.
Wouldn't the Poincare upper half-plane (or Poincare disk) work? It has constant curvature (hence Ricci curvature) $-1$ and is simply connected. All the balls with radii 1 are isometric, since the space is homogeneous, so their volumes are equal to a constant.
The volume can be computed to be $4\pi\sinh^2\frac{1}{2}$, which corresponds to that on the sphere $S^2$, a geodesic ball of radius 1 has volume $4\pi\sin^2\frac 1 2$.