Compactness of isospectral compact manifolds with bounded curvatures

Gengqiang Zhou

Vol. 181 (1997), No. 1, 187–200

DOI: 10.2140/pjm.1997.181.187

Abstract

Suppose that ℐⁿ(C) is the class of all Riemannian metrics on a given n-dimensional closed manifold such that their associated Laplacians (on functions) have the same spectrum by counting multiplicities and their sectional curvatures are uniformly bounded |K|≤ C by a constant C > 0. We show that the isospectral class ℐⁿ(C) is compact in the C^∞-topology. This generalizes our previous C^∞-compactness result, which holds for dimensions up to seven.

Milestones

Received: 1 October 1993

Revised: 18 November 1994

Published: 1 November 1997