Compact isospectral sets of plane domains
Brad G. Osgood, Ralph S. Phillips, Peter Sarnak · Proceedings of the National Academy of Sciences · 1988
Any isospectral family of two-dimensional Euclidean domains is shown to be compact in the C(infinity) topology. Previously Melrose, using heat invariants, was able to establish the C(infinity) compactness of the curvature of the boundary curves. The additional ingredient used in this paper to obtain the compactness of the domains is the behavior of the determinant of the Laplacian near the boundary of the moduli space.