Compactness of isospectral conformal metrics on \(S^ 3\)
Two Riemannian metrics g and g' on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum; isospectral metrics need not be isometric. One is interested in studying whether families of isospectral metrics are compact modulo gauge equivalence; it is natural to study this question within a conformal class. The solution of the Yamabe problem shows every conformal class of metrics on a compact Riemannian manifold contains a metric of constant scalar curvature c. \textit{R. Brooks}, \textit{P. Perry}, and \textit{P. Yang} [Duke Math. J. 58, 131-150 (1989; Zbl 0667.53037)] showed a family of isospectral metrics contained in a conformal class on a compact 3-manifold with \(c<0\) is compact in the \(C^{\infty}\) topology modulo gauge equivalence. In the present paper, the authors study this question for \(c>0\). They show a family of isospectral metrics within the conformal class of the standard metric on \(S^ 3\) is compact in the \(C^{\infty}\) topology modulo gauge equivalence; the conformal group is noncompact in this instance. In a subsequent paper [Isospectral conformal metrics on 3 manifolds (to appear)], the authors combine this result of BPY to show a family of isospectral metrics within an arbitrary conformal class on a 3- manifold is compact in the \(C^{\infty}\) topology modulo gauge equivalence. \textit{Brooks} and \textit{Gordon} [Isospectral families of conformally equivalent Riemannian metrics (to appear)] have recently shown there exist non trivial 1-parameter families of isospectral conformally equivalent metrics on \(S^ 3\times S^ 3\times {\mathbb{R}}^ 5/{\mathbb{Z}}^ 5\); it is not known whether such a deformation exists in dimension 3.
scientific article; zbMATH DE number 4113454