Note that hyperbolic space $H$ has $Ric=-(n-1)$. I want to know whether we can perturb the space so that $Ric=-(n-1)$ on $H-K$ and $Ric > -(n-1)$ for some point in $K$ where $K$ is a compact. Thank you for your attention (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)