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Moduli space, heights and isospectral sets of plane domains. (English) Zbl 0677.58045

The authors’ abstract: “Let \(\Omega\) be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain \(\Sigma\) of the same type. Then there exists a flat metric g on \(\Sigma\) such that \(\Omega\) is isometric with \(\Sigma_ g\). In what follows we do not distinguish between isometric domains. By the spectrum of \(\Sigma_ g\) we mean the spectrum of the Laplace-Beltrami operator \(\Delta_ g\) on \(\Sigma_ g\) with Dirichlet boundary conditions. The height \(h(\Sigma_ g)=-\log \det \Delta_ g\) is a spectral invariant and plays a central role in this paper. Among all suitably normalized flat metrics on \(\Sigma\) conformal to a given metric g there is a unique flat metric for which the height is a minimum. This metric is characterized by the fact that \(\partial \Sigma_ g\) has constant geodesic curvature; we call such a metric uniform and denote it by u. The set of all such metrics is denoted by \({\mathcal M}_ u(\Sigma)\). We can therefore identify \({\mathcal M}_ u(\Sigma)\) with the moduli space \({\mathcal M}(\Sigma)\) of conformal structures on \(\Sigma\). For \(n\geq 3\) we introduce a special parametrization for \({\mathcal M}_ u(\Sigma)\) by means of which we show that h(u)\(\to \infty\) as u approaches the boundary of \({\mathcal M}_ u(\Sigma)\). Using this along with the heat invariants for the Laplacian we then show that any isospectral set of plane domains is compact in the \(C^{\infty}\) topology. Similar results hold for \(n=1\) and 2.”
Reviewer: G.Warnecke

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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