Note that hyperbolic space $H$ has $Ric=-(n-1)$. I want to know :

Question : Does there exists a complete Riemannian manifold $M$ s.t.

(1) $ Ric\geq -(n-1)$ on $M$

(2) $ Ric =-(n-1)$ on $M-C$ where $C$ is a compact subset

(3) $Ric > -(n-1)$ at some point.

To construct this manifold, first we think $H$. In $H$, can we obtain a manifold satisfying these condition after pertubation ? Or in $H$ can we obtain through other way ?

Or is there manifold satisfying these condition ?

motivation : (Perelman says that if sectional curvature is nonnegative on $\mathbb{R}^n$ and some point has positive sectional curvature then it has positive sectional curvature at all points. So I ask similar question)