Large eigenvalues and concentration (Q627396)

It is a classical theme to understand relations between the geometry and eigenvalues of Riemannian manifolds. For a closed Riemannian manifold with a lower bound on the Ricci curvature, a well-known result of Cheng on an explicit upper bound on the eigenvalues in terms of the Ricci bound and the dimension implies that if the nonzero eigenvalues are very large, then the diameter of the manifold is small, i.e., the manifold is contained in a small neighborhood of any point of the manifold. One of the major results of this paper is the following. Suppose that a compact Riemannian manifold \(M\) satisfies the covering property that the number of balls of radius \(r\) needed to cover a ball of radius \(4r\) is uniformly bounded above by a common constant for all \(r\leq 1\). If the \((k+1)\)-th eigenvalue of \(M\) is large, then most of the volume of \(M\) concentrates near at most \(k\) points of the manifold. A more general result holds for eigenvalues of a Laplace-type operator acting on sections of a smooth vector bundle on \(M\).