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Resolving decomposition by blowing up points and quasiconformal harmonic extensions : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
In this thesis we consider two problems regarding mappings between various two-dimensional
spaces with some constraint on their distortion.
The first question concerns the use of mappings of finite distortion that blow up
a point where the distortion is in some Lp class; in particular, we are interested in
minimal solutions to the appropriate functional. We first prove some results concerning
these minimal solutions for a given radially symmetric metric (in particular
the Euclidean and hyperbolic metrics) by proving a theorem which states the conditions
under which a minimizer exists, as well as providing lower bounds on the
Lp-norm of the function. We then apply these results to the problem of resolving
decompositions that arise in the study of Kleinian groups and the iteration of rational
maps. Here we prove a result concerning for which values of p we can find a
mapping of a particular form which shrinks the unit interval and whose inverse has
distortion in the Lp space.
The second is in regards to the Schoen conjecture, which expresses the hope that
every quasisymmetric self-mapping of the unit circle extends to a homeomorphism of
the disk which is both quasiconformal and harmonic with respect to the hyperbolic
metric. The equation for a harmonic map between Riemann surfaces with given
conformal structures is a nonlinear second order equation; one wishes to solve the
associated boundary value problem. We show here that the existence question can
be related to a nonlinear inhomogeneous Beltrami equation and discuss some of the
consequences; this result holds in more generality for other conformal metrics as