Local boundary dilatation of quasiconformal maps in the disk (Q2781296)

In this paper the authors prove a result conjectured by \textit{F. Gardiner} and \textit{N. Lakic} in ``Quasiconformal Teichmüller theory'' (2000; Zbl 0949.30002). This states that if \(\Omega\) is a planar domain and let \(\zeta\) be a boundary point of \(\Omega\). Denote by \(T(\Omega)\) the Teichmüller space of \(\Omega\) which we consider as classes of Beltrami coefficients. Let \(\tau \in T(\Omega)\). Then the assertion is that there is a \(\mu\) representing \(\tau\) so that the dilation of \(\mu\) in an infinitesimal neighbourhood of \(\zeta\) (boundary dilation of \(\mu)\) achieves the infinum over all Beltrami admissible Beltrami coefficients. The proof is based on a localization/gluing argument which uses a comparison theorem for quadratic differentials due to \textit{E. Reich} and \textit{K. Strebel}, Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers 375-391 (1974; Zbl 0318.30022).