Abstract
This chapter presents a geometric method to extend a quasiconformal mapping from Rn to Rn+1. For n = 1, the problem was solved by Ahlfors and Beurling, through an ingenious explicit integral formula. Ahlfors proved the result for n = 2. It seems quite difficult to generalize the factorization to n ≥ 3 and therefore, the chapter presents a method that is geometric and constructive. However, it depends heavily on approximations of arbitrary homeomorphisms by piecewise linear mappings, and it requires quite detailed information about the possibility of fitting together such mappings.
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