On the extremality of quasiconformal mappings and quasiconformal deformations

摘要

Given a family of quasiconformal deformations F(w,t) such that <(partial derivative)over bar>F has a uniform bound M, the solution f(z, t)(f(z, 0) = z) of the Lowner-type differential equation dw/dt = F(w, t) is an e(2Mt)-quasiconformal mapping. An open question is to determine, for each fixed t >0, whether the extremality of f(z, t) is equivalent to that of F(w, t). The note gives this a negative approach in both directions. [References: 9]