Compactness of isospectral compact manifolds with bounded curvatures (Q1816514)

Suppose that \({\mathfrak J}^n (C)\) 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,\dots, by a constant \(C>0\). We show that the isospectral class \({\mathfrak J}^n (C)\) is compact in the \(C^\infty\)-topology. This generalizes our previous \(C^\infty\)-compactness result, which holds for dimensions up to seven.