Geodesic
Quasiconformal homotopies of elementary space mappings
Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 205-212

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This article takes up the problem of a quasiconformal homotopy to the identity quasiconformal space mapping for the model case of an elementary piecewise-affine mapping of a simplex. In view here are continuous orientation-preserving mappings of the simplex that are affine on its boundary and in each simplex of the decomposition obtained by adding a single new vertex inside the original simplex. It is proved that an arbitrary elementary piecewise-affine mapping of the simplex admits a quasiconformal homotopy to the identity mapping. The proof is based on the following assertion: the smallest coefficient of quasiconformality in the class of all elementary piecewise-affine mappings of the simplex that coincide on its boundary with some affine mapping belongs to this affine mapping. This result can be regarded as a multidimensional analogue of the classical Grötzsch problem on an extremal mapping of rectangles that deviates least from a conformal mapping. Bibliography: 4 titles. 
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     title = {Quasiconformal homotopies of elementary space mappings},
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I. V. Abramov; E. A. Roganov. Quasiconformal homotopies of elementary space mappings. Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 205-212. http://geodesic.mathdoc.fr/item/SM_1991_68_1_a10/