The main goal of this book is to present the basic tools that are necessary for research in geometric analysis. Though the main theme centers around linear theory, i.e., the Laplace equation, the heat equation, and eigenvalues for the Laplacian, the methods of dealing with these problems are quite often useful in the study of nonlinear partial differential equations that arise in geometry.

A small portion of this book originated from a series of lectures given by the author at a Geometry Summer Program in 1990 at the Mathematical Sciences Research Institute in Berkeley. The lecture notes were revised and expanded when the author taught a regular course in geometric analysis. During the author's visit to the Global Analysis Research Institute at Seoul National University, he was encouraged to submit these notes, though still in a rather crude form, for publication in their lecture notes series [L6].

The part of this book that concerns harmonic functions originated from the author's lecture notes for a series of courses he gave on the subject at the University of California, Irvine. A part of this material was also used in a series of lectures the author gave at the XIV Escola de Geometria Diferencial in Brazil during the summer of 2006. These notes [L9] were printed for distribution to the participants of the program.