A fundamental variational lemma for extremal quasiconformal mappings compatible with a Fuchsian group (Q1820271)

In this paper the author carries over the fundamental variational proved by \textit{R. Fehlmann} in Comment. Math. Helv. 61, 565-580 (1986) to the setting of Fuchsian groups \(\Gamma\). The lemma enables one to replace an infinitesimally trivial Beltrami differential by an equivalent Beltrami differential with much smaller norm. This replacement is accomplished with the side condition that the Beltrami coefficients be identically equal to zero on a specified closed \(\Gamma\)-invariant subset E of the unit disk. The set E is assumed to be measurable and to have a boundary of measure zero. It is also assumed to satisfy a relative compactness condition on the quotient surface equal to the unit disk factored by the Fuchsian group. The purpose of this lemma is to enable one to extend the results of \textit{R. Fehlmann} [Ann. Acad. Sci. Fenn., Ser. A I 7, 337-347 (1982; Zbl 0487.30008)], \textit{F. P. Gardiner} [Mich. Math. J. 29, 237-242 (1982; Zbl 0476.30015)] and \textit{K. Sakan} [J. Math. Kyoto Univ. 26, 31- 37 (1986; Zbl 0604.30023)] to Riemann surfaces and to infinite dimensional Teichmüller spaces.