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

 Question : Does there exists a simply connected open 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 : (1) 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

(2) That is, if Ricci curvature condition on $M$, which is diffeomorphic to $H$, is slightly different from $H$, in fact, $M$ is isometric to $H$ ? Thank you for your attention.