Eigenvalues of elliptic operators and geometric applications

The purpose of this talk is to present a certain method of obtaining upper estimates of eigenvalues of Schrodinger type operators on Riemannian manifolds, which was introduced in the paper Grigor’yan A., Netrusov Yu., Yau S.-T. Eigenvalues of elliptic operators and geometric applications, Surveys in Differential Geometry IX (2004), pp.147-218. The core of the method is the construction that allows to choose a given number of disjoint sets on a manifold such that one can control simultaneously their volumes from below and their capacities from above. The main technical tool for that is the following theorem. Theorem 1 Let (X, d) be a metric space satisfying the following covering property: there exists a constant N such that any metric ball of radius r in X can be covered by at most N balls of radii r/2. Let all metric balls in X be precompact sets, and let ν be a non-atomic Radon measure on X. Then, for any positive integer k, there exists a sequence {Ai}ki=1 of k annuli in X such that the annuli {2Ai}ki=1 are disjoint and, for any i = 1, 2, ..., k