Chapter 3 The Laplace Operator and Harmonic Differential Forms

Abstract

A fundamental topic in geometric analysis and an important tool for studying Riemannian manifolds is given by harmonic objects. Such objects are defined as the minimizers, or more generally, the critical points of some action or energy functional. We have already seen one instance, the energy functional for curves in a Riemannian manifold, see (1.4.7), (1.4.9), whose critical points were the geodesics, see (1.4.14). In this chapter, however, we shall not look at maps into a Riemannian manifold, but at functions and differential forms defined on such a manifold. (These two themes, maps into a manifold and functions on a manifold, will be unified in Chapter 8 below.) In this section, which is of an introductory nature, we consider harmonic functions on Riemannian manifolds. More generally, we discuss the energy functional, the Dirichlet integral, and the operator, the Laplace–Beltrami operator, in terms of which harmonic functions are defined. This functional and operator will play an important role at many places in this book. Since the emphasis in this section will be on introducing concepts, we shall not go into the analytical details. Those will be presented in Appendices A.1 and A.2.