On Fredholm eigenvalues of unbounded polygons (Q2282620)

Suppose that \(P\) is a Jordan curve in the Riemann sphere. One can consider various complex analytic functionals related to \(P\): the first Fredholm eigenvalue (defined, e.g., using an extremal problem involving Dirichlet integrals), the Grunsky norm (defined using logarithmic coefficients of a Riemann map for the exterior of \(P\)) and the Teichmüller norm (the minimal dilatation among quasiconformal extensions of a Riemann map). The problem of the determination of these functionals and their interrelations is, in general, difficult and open. The author studies the case when \(P\) is a polygon and shows that in a special case all these functionals coincide.