Isoperimetric inequalities and the first eigenvalue of the Laplacian on a Riemannian manifold.

摘要

In this survey, we will study the relationship between isoperimetric inequalities and the first eigenvalue of the Laplacian on a Riemannian manifold. We will take two approaches. The Faber-Krahn inequality shows that a certain isoperimetric comparison theorem implies an eigenvalue comparison theorem. Cheeger's inequality gives a lower bound for the first eigenvalue in terms of an isoperimetric constant h(M). As well as attempting to understand the relationship between h( M) and the first eigenvalue, we will calculate h( M) for certain special cases and get useful approximations for more general situations.