Moduli Space, Heights and Isospectral Sets of Plane Domains

Abstract

Let Q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type. Then there exists a flat metric g on 2 such that Q is isometric with 29g. In what follows we do not distinguish between isometric domains. By the spectrum of 29 we mean the spectrum of the Laplace-Beltrami operator Ag on 29 with Dirichlet boundary conditions. The height h(Eg) = - log det Ag is a spectral invariant and plays a central role in this paper. Among all suitably normalized flat metrics on 2 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 d 2 has constant geodesic curvature; we call such a metric uniform and denote it by u. The set of all such metrics is denoted by u(2). We can therefore identify ( with the moduli space #(2) of conformal structures on E. For n ? 3 we introduce a special parametrization for Mu(2) by means of which we show that h(u) - oo as u approaches the boundary of fu(2). 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' topology. Similar results hold for n = 1 and 2.