Compactness of isospectral conformal metrics onS 3

Abstract

Two Riemannian metrics g, g' on a compact manifold are said to be isospectral if their associated Laplacian operator on functions have identical spectrum. It is a well known problem to study the extent to which the spectrum determines the metric. In dimension two, Osgood, Phillips and Sarnak [OPS] studied this question and proved that the set of isospectral metrics on a compact surface form a compact family in the cr topology. In that case there is available a criterion due to Wolpert [W] for compactness of the conformal structures in the Teichmuller space in terms of the determinant of the Laplacian. This reduces the problem to studying the isospectral conformal metrics on a fixed Riemann surface. It turns out that the determinant of the Laplacian played the key role for the compactness questions. In particular when the underlying surface is the two sphere, which is analytically the least transparent case, the compactness question reduces to an inequality of Onofri ([O], [OPS]) which is a sharp version of the Moser-Trudinger inequality on S 2. We are interested in the situation in dimension 3. The well known solution of the Yamabe problem ([A], [S]) says that every conformal class of metrics on a compact Riemannian manifold contains a metric of constant scalar curvature. When (M 3, go) has constant negative scalar curvature, an isospectral set of metrics g = u4go conformal to go is compact in the cr174 topology [BPY]. This result was proved directly using the heat invariants of the metric. The first step was to find a pointwise bound 0< cl- u-< c2 and a bound 1tu112.2 <-c3 where ci depend only on the heat invariants of g. The higher order derivative bounds required for cr174 compactness is a consequence of this bound for u and the calculation for the coefficients for the terms involving the highest order derivatives of u in the asymptotic ak of the heat kernel for g due to Gilkey ([G]). In this paper we study the situation when M is the standard three sphere ( $3, go). As in the case of the two sphere, the conformal group G complicates the analysis.