On the extremality of quasiconformal mappings and quasiconformal deformations

Abstract

Given a family of quasiconformal deformations F (w, t) such that ∂F has a uniform bound M , the solution f(z, t)(f(z, 0) = z) of the Lownertype differential equation dw dt = F (w, t) is an e2Mt-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. §