Upper Bounds for Eigen values of Laplace operator on manifold

Abstract

In this paper, we are concerned with upper bounds of Eigen values on manifolds. Eigen values have many applications in geometry and in other fields of mathematics. We develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel. we start with a well-defined Laplace operator ? on functions on M so that ? is a self-ad joint operator in L2(M, +) with a discrete spectrum and a distance function dist(x, y) on M.
Keywords: Eigen values, Laplace transform, heat equation, manifold