Quasiconformal mappings on planar surfaces

This thesis discusses three different projects concerning quasiconformal mappings on planar surfaces. In the first two projects we show that a priori weaker conditions still suffice to prove quasiconformality. The geometric definition states that an orientation-preserving homeomorphism f:U → f(U) is quasiconformal if there exists K ≥ 1 such that for all Q, Q ⊂ U the ratio of the modulus of f(Q) to the modulus of Q is bounded above by K. We show that for the subclass of homeomorphisms that preserves the set of vertical lines, it suffices to just consider squares with sides parallel to the coordinate axes and at forty-five degree angles to the coordinate axes. Another more recent sufficient condition for quasiconformality discovered by Hubbard in 2006, requires that the skews of triangles be only distorted by a bounded amount. Haïssinsky, Hinkkanen and I proved that if there exists a constant K such that (f(T)) ≤ K for all equilateral triangles T, then f is quasiconformal. Furthermore, this condition is also sufficient for mappings between finite-dimensional Hilbert spaces. In the last project we study quasiconformal mappings on a generalized class of Grushin planes. We define quasisymmetries between these Grushin planes and the complex plane, and use them to find a Grushin Beltrami equation and state an analytic definition of quasisymmetry on the Grushin plane. Finally we look at a previously discovered class of conformal mappings on the Grushin plane, and show that these conformal mappings agree with our analytic definition of quasisymmetry in the conformal case