In honor of Richard Schoen for the occasion of his 60th birthday STRUCTURE OF COMPLETE MANIFOLDS WITH POSITIVE SPECTRUM

In this article, we will give a brief survey on some recent development concerning the understanding of the structure at infinity of a complete manifold whose spectrum has a positive lower bound. Throughout this paper, we denote M to be a complete n-dimensional manifold without boundary endowed with the metric ds. We assume that the Ricci curvature of M is bounded from below by some constant. Recall that when M has nonnegative Ricci curvature, then the splitting theorem of Cheeger and Gromoll [CG] asserts that M must have only one end unless M is a cylinder isometric to the product, R × N , of the real line and a compact manifold Nn−1 with nonnegative Ricci curvature. For the purpose of this discussion, we may hence assume that the Ricci curvature of M is bounded from below by a negative constant, and after rescaling,