Hamilton sequences for extremal quasiconformal mappings in the unit disk (Q1974217)

A quasisymmetric self map \(h\) of the circle is given and it is assumed that \(h\) has more than one extremal quasiconformal extension, say \(f_1\) and \(f_2\), with Beltrami coefficients \(\mu_1\) and \(\mu_2\), and each with extremal dilatation \(K=K(h)\). A sequence of quadratic differentials \(\varphi_n\) defined in the disc \(\Delta\) with norm 1 is called a Hamiltonian sequence for \(\mu_1\) if \[ \iint_\Delta\varphi_n\mu_1 dxdy\to \|\mu_1\|_\infty, \] and the sequence \(\varphi_n\) is called a maximal norm sequence for \(h\) if \(\|\varphi_n(h)\|\to K\), where \(\varphi_n(h)\) is the image under the heights mapping by \(h\) of \(\varphi_n\), defined by \textit{K. Strebel} [Ann. Acad. Sci. Fenn., Ser. AI 18, No. 1, 155-190 (1993; Zbl 0783.30019)]. The author shows that if \(\varphi_n\) is a Hamiltonian sequence for \(f_1\) then it is a maximal norm sequence for \(h\), and also that if \(\varphi_n\) is a maximal norm sequence for \(h\), then it is a Hamilton sequence for any extremal representative of \(h\). In particular, any Hamilton sequence for \(\mu_1\) must also be a Hamilton sequence for \(f_2\). Finally, he shows that if \(\varphi_n\) is a maximal norm sequence for \(h\), then \(-\frac{\varphi_n(h)}{\|\varphi_n(h)\|}\) is a maximal norm sequence for \(h^{-1}\). He uses techniques similar to those used in \textit{N. Lakic} [Mich. Math. J. 44, No. 2, 209-316 (1997); \textit{E. Reich}, and \textit{K. Strebel}, Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers., 375-391 (1974; Zbl 0318.30022)].