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 $H$, satisfies the above conditions, in fact, $M$ is isometric to $H$ ? Thank you for your attention.

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    $\begingroup$ What about the half-cylinder with a half-sphere attached? It has a point with K>0 and other points with K=0, and is diffeomorphic to R^2? $\endgroup$ – valeri May 9 '15 at 16:18
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    $\begingroup$ Do you care if the metric changes on $H-K$ as long as the Ricci curvature stays fixed? $\endgroup$ – Otis Chodosh May 9 '15 at 17:18
  • $\begingroup$ My asking is vogue I will edit it. $\endgroup$ – Hee Kwon Lee May 9 '15 at 21:25
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    $\begingroup$ Note, that changing Ricci to sectional curvature for n>2 the answer to your question is negative: projecteuclid.org/euclid.hokmj/1381413169 $\endgroup$ – valeri May 10 '15 at 10:37
  • $\begingroup$ Thank you for your introducing the paper. It is interesting to me $\endgroup$ – Hee Kwon Lee Mar 23 '16 at 6:39

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