Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

摘要

Let M be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on M to be extremal for lambda(1) with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice Gamma of R-n, the flat metric g(Gamma), induced on R-n/Gamma from the standard metric of R-n is extremal (in the previous sense). In the second part, we give, for any Gamma, an upper bound of lambda(1) on the conformal class of g(Gamma), and exhibit a class of lattices Gamma for which the metric g(Gamma), maximizes lambda(1) on its conformal class. [References: 12]