Finite spectral determination of compact homogeneous manifolds

A finite initial segment of a prescribed scalar Laplace spectrum determines the full spectrum among closed homogeneous manifolds of the same dimension. The realizations are represented by finitely many compact semialgebraic sets of positive invariant cometrics. They are therefore smoothly compact modulo diffeomorphisms and have finitely many diffeomorphism types. Among globally naturally reductive metrics, each spectrum has only finitely many isometry classes. For these metrics, a finite set of scalar eigenvalues also gives a local Lipschitz estimate for the metric modulo diffeomorphisms, simultaneously in every differentiability order. An explicit family on SU(2) × S¹ has fixed volume and uniform relative convergence of all positive eigenvalues, while its scalar curvature tends to −∞. Zenodo publication date: 1 October 2026. Source: version v2, DOI 10.5281/zenodo.23307538.