L(?)wner Differential Equation And The Universal Teichmüller Space

We apply the Lowner differential equation to discussing the parametric representations of quasiconformal mappings and quasisymmet-ric homeomorphisms. Roughly speaking, when F(w,t) are quasiconformal deformations, the solutions f(z,t) are quasiconformal mappings; when F(w,t) are Zygmund functions, the so-lutions f(z,t) are quasisymmetric homeomorphisms. In this paper, we study further the relations between the vector fields F(w,t) and the solutions f(z,t). In particular, we prove that the quasisymmetric homeomorphism solutions to the Lowner differential equation for the vector field of "smooth" Zygmund functions belong to the small Teichmuller space. Fur-thermore, such solutions can generate the whole small Teichmuller space.