= Compact isospectral sets of closed surfaces
For a fixed <closed> orientable <Riemannian surface>, the smooth metrics with a specified scalar <Laplace-Beltrami operator> spectrum form a <compact> set modulo <diffeomorphisms>, in the smooth topology. This is a <compactness> assertion, not uniqueness or finiteness of the number of <Riemannian isometry> classes. It restricts geometric degeneration within an isospectral set and uses more information than the leading <heat invariants> alone. The theorem was established in https://doi.org/10.1016/0022-1236(88)90071-7[the 1988 compactness paper].
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