A gap theorem for ends of complete manifolds

Abstract:

Let $({M^n},o)$ be a pointed open complete manifold with Ricci curvature bounded from below by $- (n - 1){\Lambda ^2}$ (for $\Lambda \geq 0$) and nonnegative outside the ball $B(o,a)$. It has recently been shown that there is an upper bound for the number of ends of such a manifold which depends only on $\Lambda a$ and the dimension n of the manifold ${M^n}$. We will give a gap theorem in this paper which shows that there exists an $\varepsilon = \varepsilon (n) > 0$ such that ${M^n}$ has at most two ends if $\Lambda a \leq \varepsilon (n)$. We also give examples to show that, in dimension $n \geq 4$, such manifolds in general do not carry any complete metric with nonnegative Ricci Curvature for any $\Lambda a > 0$.