Spectral Theory and Geometry
Bruno Colbois · 2009
Preamble : These are informal notes of a series of 4 talks I gave in Teheran, in the CIMPA-UNESCO-IRAN School ”Recent Topics in Geometric Analysis”, May 21-June 2, 2006. The goal was to give an introduction to the geometric spectral theory of the Laplacian acting on p-differential forms. Of course, in a few hours, it is hopeless to be complete, and I had to make some choice for the content of these lectures. Clearly, the main purpose is to give an intuition based on examples about the following question : to what extend is it possible to construct large or small first nonzero eigenvalue for the Laplacian on forms on a compact Riemannian manifold. This point of view is quit reductive, in particular I do not say one word about the asymptotic of the spectrum and about the heat kernel, and the reader interested on such aspects of the theory may look at the book of Rosenberg [Ro]. However, this question of large or small eigenvalues is easy to understand, it allows to make clear the importance of the topology of the manifold, much more crucial as in the case of functions, and also give the opportunity to present a lot of open but accessible questions. In the introduction (first lecture), I just recall without many explanations the basis of the Laplace operator and of its spectrum. We do it first for functions and then in the case of p-forms. This is of course partially redundant, but I expected the audience more or less familiar with the case of functions and not necessarily with the case of forms. There exists among other three excellent monographs about this question : the very complete book of M. Taylor [Ta] and the lecture note of G. Schwarz [Sc], mainly for the questions related to the boundary conditions, and the above mentioned book of Rosenberg [Ro], mainly concerned with the asymptotic of the spectrum, and whose introduction is very instructive. We also recall the De Rham theory and Hodge decomposition for compact Riemannian manifolds. In the beginning, in the case of functions, I will mainly refer to the books of P. Berard [Be] and I. Chavel [Ch]. I profited of the introduction to recall some typical and classical results in the case of functions, because what is known in this context is inspiring of what we try to do for forms.