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    The generic failure of lower-semicontinuity for the linear distortion functional : a thesis presented in partial fulfilment of the requirments for the degree of Doctor of Philosophy (PhD) in Mathematics, Institute of Advance Studies (NZIAS) Massey University of Auckland, New Zealand
    (Massey University, 2022) Hashemi, Seyed Mohsen
    My research is primarily concerned with the convexity properties of distortion functionals (particularly the linear distortion) defined for quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal application is in studying the lower semi-continuity of distortion on uniformly convergent limits of sequences of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality - discussed in Chapter 3 - it is known that if {fₙ}∞ₙ₌₁ is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function f, then the limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (via the linear distortion H(f) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this thesis is to show that this failure of lower semicontinuity is actually far more common, perhaps even generic in the sense that it is true that under mild restrictions on a quasiconformal f, there may be a sequence {fₙ}∞ₙ₌₁ with fₙ → f uniformly and with limsupₙ→∞ H(fₙ) < H(f). The main result of this thesis is to show this is true for a wide class of linear mappings and give bounds for the maximal jump down in the limit.
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    Minimisation of mean exponential distortions and Teichmüller theory : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2019) Yao, Cong
    This thesis studies the Cauchy boundary value problem of minimising exponential integral averages of mappings of finite distortion. Direct methods in calculus of variations provide existence theorems and we derive the Euler-Lagrange equations for minimisers of ∫D exp(pK(z, f)) dz for mappings of finite distortion f : D → D with prescribed boundary values. However, surprisingly, for these functionals some apriori regularity is needed before we can discuss these equations. We show by example how this can happen. We construct a mapping f : D → D with exponentially integrable distortion to exponent p which cannot perturbed by any diffeomorphism and still remain exponentially integrable with exponent p. Once enough apriori regularity is assumed for instance if a minimiser is locally quasiconformal, that is if the distortion function K(z, f) is locally bounded, then we use these equations to improve the regularity of the minimisers. In particular, we find that minimisers with locally bounded distortions are diffeomorphisms. Then we analyse the two extreme cases (1) p → 0 and (2) p → ∞. In this way we see the p-exponential problem connects the L¹ finite distortion problem, which is closely related to the classical harmonic theory in case (1), and to the Teichmüller problem, which promoted the development of quasiconformal mappings, in case (2).
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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
    (Massey University, 2012) Dillon, Samuel Adam Kuakini
    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 well.