On compactness of isospectral conformal, metrics of 4-manifolds

Abstract

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metricsg, g′on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operatorΔon the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).