Some Analytical Properties For Harmonic Quasiconformal Mappings

In order to derive more information on harmonic mappings from Riemann manifolds,Yau said that one often needs to assume that it is also quasiconfor-mal.Recently,Iwaniec,Kovalev and Onninen studied harmonic quasiconformal mappings between two finitely circular annuli A and A1,and used the obtained results to solve Bjorling problem in minimal surfaces.Basic on the ideas to in-vestigate mappings by combining harmony with quasiconformality,in this thesis we mainly study several analytical properties for planar harmonic quasiconformal mappings.It mainly splits into 3 parts:establishing Heinz type inequalities for mappings satisfying Poisson’s equation;giving the explicit upper and lower esti-mates of hyperbolically partial derivative for quasiconformal mappings satisfying Poisson’s equation;giving a rigidity result on simple closed curves on closed surface with genusg>1.In Chapter 1,we mainly introduce some research backgrounds,current re-search situations,research contents and methods,and main results for planar harmonic quasiconformal mappings and the homotopy of closed curves on closed surface with genus g">1.In Chapter 2,we discuss the Heinz type inequalities for mappings satisfying Poisson’s equation from two cases.In first case,we studied the Heinz type in-equality for a twice continuously differentiable function ω from unit disk D onto itself satisfying Poisson’s equation,where its boundary valve function is a sense-preserving homomorphism of unit circle.In another case,by using generalized Heinz inequality,we gaved Heinz type inequalities for K-quasiconformal mappings of unit disk D onto itself satisfying Poisson,s equation △ω g,g ∈(?)C(D).The obtained results partially generalized some classical results on planar harmonic quasiconformal mappings derived by Partyka and Sakan.In Chapter 3,we discuss the explicit upper and lower estimates of hyperboli-cally partial derivative for quasiconformal mappings satisfying Poisson’s equation.Namely,let ω be a K-quasiconformal mapping from unit disk D onto itself satisfy-ing △ω= g,g ∈C(D).By establishing the generalized Heinz inequality for univa-lent harmonic mapping of unit disk D onto itself,we obtained explicit upper and lower estimates of hyperbolically partial derivative for ω.Moreover,we proved that these estimations are asymptotically sharp in the sense that ||g||∞ = supz∈D|a(z)|tends to zero.As applications,hyperbolic area distortion and Euclidean length distortion estimate are obtained and we proved that these area distortions are asymptotically sharp in the sense that K→1+ and ||g||∞ ends to zero.In Chapter 4,we mainly give a rigidity result on simple closed curve on closed surface.Namely,let S be a close surface with genus g≥2 and let γ1,γ2 be two simple closed curves on S.By virtue of the height of simple loop and Jenkins-Strebel differential,we proved that:if the q-height of γ1,γ2 are the same for any quadratic differential q in S with ||q|| = 1,then γ1,γ2 represent the same homotopy class.