Spectral geometry in dimension 3
The authors study the question: to what extent does the spectrum of the Laplacian of a closed manifold \(M\) determine its geometry and topology? The curvature tensor of a Riemannian manifold has a pointwise decomposition in the form \(R(M) = S \oplus \widetilde {Ricc} \oplus W\) where \(S\), \(S \widetilde {Ricc}\), and \(W\) are the scalar, traceless Ricci, and Weyl tensor. Let \(| Ricc |_{2,red}\) be the \(L^ 2\) norm of the reduced curvature tensor which is defined by \(R(M)_{red} = \{S - \text{vol} (M)^{-1} \cdot \textstyle \int_ MS\} \oplus \widetilde {Ricc} \oplus W\). Let \(\lambda_ k\) denote the \(k^{th}\) eigenvalue of the Laplacian. The authors show Theorem: In dimensions 2 or 3 and for any positive integer \(k\), there are constants \(Q(n)\) and \(K(n,k)\) so that if \(\lambda_ k > Q(n) \text{vol} (M)^{-1} \int_ MS\) and if \(\lambda_ k > K(n,k) \text{vol} (M)^{- 1/2} | Ricc |_{2,red}\), then the set of manifolds which carry metrics isospectral to \(M\) contains only finitely many diffeomorphism types and the set of such metrics is compact in the \(C^ \infty\) topology. It then follows as a corollary that there are spectrally determined open sets about manifolds of constant curvature for which one has compactness of isospectral sets of metrics in dimension 3.
scientific article; zbMATH DE number 719523