A GAP THEOREM FOR ENDS OF COMPLETE MANIFOLDS

Let (M(n), o) be a pointed open complete manifold with Ricci curvature bounded from below by -(n - 1)Lambda(2) (for Lambda greater than or e qual to 0) and nonnegative outside the ball B(o, a). It has recently b een 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 m anifold M(n). We will give a gap theorem in this paper which shows tha t there exists an epsilon = epsilon(n) > 0 such that M(n) has at most two ends if Lambda a less than or equal to epsilon(n). We also give ex amples to show that, in dimension n greater than or equal to 4, such m anifolds in general do not carry any complete metric with nonnegative Ricci Curvature for any Lambda a > 0.