Analysis of the Laplace operator on manifolds

We discuss the Laplacian on complete noncompact Riemannian manifolds. We provide a proof of the Hessian Comparison Theorem which does not depend on Jacobi fields or matrix Ricatti equations. We present a unified treatment of three important known facts for complete noncompact Riemannian manifolds: the essential self-adjointness of the Laplace operator; the nonexistence of {dollar}Lsp2{dollar} harmonic functions; and the identity {dollar}intvertbigtriangledown uvertsp2=lambdaint usp2{dollar} for {dollar}Lsp2{dollar} eigenfunctions with eigenvalue {dollar}lambda.{dollar} For manifolds with nonegative Ricci curvature, we prove an identity which shows in particular the square integrability of the Hessian of an eigenfunction. We provide a detailed proof of the decomposition principle for the essential spectrum. We review the spectrum of the one-dimensional Schrodinger operator and use it to prove theorems on the spectrum of warped products. We establish a connection between the essential spectrum and Rellich-type properties. Finally we study perturbations of the Laplacian by a vector field. We obtain gradient estimates for positive solutions of the equation {dollar}Delta u+Xu{dollar} where X is a vector field and obtain a Liouville property.