# Moduli space, heights and isospectral sets of plane domains

### 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 $\mathscr{M}_u(\Sigma)$. We can therefore identify $\mathscr{M}_u(\Sigma)$ with the moduli space $\mathscr{M}(\Sigma)$ of conformal structures on $\Sigma$. For $n \ge 3$ we introduce a special parametrization for $\mathscr{M}_u(\Sigma)$ by means of which we show that $h(u) \to \infty$ as $u$ approaches the boundary of $\mathscr{M}_u(\Sigma)$. 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 $2$.