Compact lsospectral Sets of Surfaces

In this paper we study sets of surfaces which are isospectral with respect to the Laplace-Beltrami operator. More specifically, for closed surfaces (compact, no boundary) we consider a fixed surface and the family of metrics on that surface having a given Laplace spectrum, whereas for surfaces with boundary we confine our study to the class of simply connected planar domains all having the same spectrum for the euclidean Laplacian with Dirichlet boundary conditions. This latter case is classical and directly tied to “hearing the shape of a drum” as the problem was posed by Kac [9]. Briefly, our main result is that such isospectral families of metrics, or plane domains, are compact in a natural C” topology. To be more precise, let A4 be a closed surface and fix an arbitrary background metric on M. With respect to this metric we can define the space of Ck metric tensors on M. Two Ck metrics are isometric if there is a Ck diffeomorphism of A4 onto itself under which the metrics correspond. Since isometric metrics have, in particular, identical Laplace spectra, what is of interest in our problem are the classes of isometric metrics on M. If g denotes the isometry class of g then the Ck topology on such classes is defined by 8, + 2 if there are metrics h, E g,, and h E 2 such that h, + h in Ck. This is described in more detail in Section 2. We shall prove that an isospectral set of isometry classes of metrics on a closed surface is sequentially compact in the C” topology. The cases of genus zero and higher genus require different treatments. Consider next simply connected planar domains with smooth boundary. By the Riemann mapping theorem we can use the unit disk U as a reference surface and parameterize such domains by flat metrics on U conformal to the euclidean metric. The topology on isometry classes of metrics