Comparing Poincaré densities. (English) Zbl 1045.30026

Let \(\Omega\) be a hyperbolic domain in the extended complex plane \(\bar{\mathbb C}\) and let \(\rho_\Omega\) denote its Poincaré (hyperbolic) density. By domain monotonicity (Schwarz lemma), \[ \rho_\Omega(z)\geq \hat{\rho}_\Omega(z):=\sup\{\rho_{\bar{\mathbb C}\setminus \{a,b,c\}}(z): a,b,c\in \bar{\mathbb C}\setminus \Omega\}. \] This leads to explicit estimates for \(\rho_\Omega\) because there exists an explicit formula for the Poincaré density of \(\bar{\mathbb C}\setminus \{a,b,c\}\). The proof is based on a new canonical density \(\lambda_\Omega\) (called Teichmüller density) which turns out to be equivalent to \(\hat{\rho}_\Omega\) and to \(\rho_\Omega\). It is defined in terms of vector fields \(V_z\) on \(\Omega\) which are equal to one at \(z\) and vanish on \(\partial \Omega\). These vector fields are tangent to holomorphic motions which initially move the point \(z\) with unit velocity while keeping the boundary fixed. The rigorous definition of \(\lambda_\Omega\) involves quadratic differentials and Teichmüller theory. Thus the equivalence of \(\lambda_\Omega\) and \(\rho_\Omega\) provides a link between geometric function theory and hyperbolic geometry on the one hand, and Teichmüller theory and holomorphic motions on the other hand. The authors give applications of the above result on uniformly perfect domains and other special classes of domains.


30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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