Quasidisks and twisting of the Riemann map

Consider a conformal map from the unit disk on to a quasidisk. We determine a range of critical complex powers with respect to which the derivative is integrable. The results fit into the picture predicted by a circular analogue of Brennan's conjecture.Peer reviewe

Quasiconformal extensions, Loewner chains, and the lambda-Lemma

Becker (J Reine Angew Math 255: 23-43, 1972) discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker's result based on Slodkowski's Extended.-Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker's (classical) construction and use that to obtain examples in which Becker's extension is extremal (i. e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.Peer reviewe

Dimensiodistortio kvasikonformikuvauksissa

Quasiconformal mappings are natural generalizations of conformal mappings. They are homeomorphisms with 'bounded distortion' of which there exist several approaches. In this work we study dimension distortion properties of quasiconformal mappings both in the plane and in higher dimensional Euclidean setting. The thesis consists of a summary and three research articles. A basic property of quasiconformal mappings is the local Hölder continuity. It has long been conjectured that this regularity holds at the Sobolev level (Gehring's higher integrabilty conjecture). Optimal regularity would also provide sharp bounds for the distortion of Hausdorff dimension. The higher integrability conjecture was solved in the plane by Astala in 1994 and it is still open in higher dimensions. Thus in the plane we have a precise description how Hausdorff dimension changes under quasiconformal deformations for general sets. The first two articles contribute to two remaining issues in the planar theory. The first one concerns distortion of more special sets, for rectifiable sets we expect improved bounds to hold. The second issue consists of understanding distortion of dimension on a finer level, namely on the level of Hausdorff measures. In the third article we study flatness properties of quasiconformal images of spheres in a quantitative way. These also lead to nontrivial bounds for their Hausdorff dimension even in the n-dimensional case.Kvasikonformikuvaukset ovat konformikuvausten yleistyksiä, homeomorfismeja joiden distortio on rajoitettu. Nämä kuvaukset muodostavat joustavan bilipschitz-kuvausten ja yleisten homeomorfismien väliin sijoittuvan kuvausluokan, jolla on useita merkittäviä geometrisiä ja analyyttisiä ominaisuuksia. Tässä työssä tutkitaan kvasikonformikuvausten dimensiodistortiota. Pisteittäiset distortiota koskevat perustulokset todistettiin jo alan kehityksen alkuvaiheissa. Dimension muuntuminen kvasikonformikuvauksissa tarjoaa haasteellisemman aihepiirin. Yleisten joukkojen tapauksessa täydellinen ratkaisu tunnetaan vain tasossa (Astala, 1994). Tällaiset metriset distortiotulokset ovat myös läheisessä yhteydessä kvasikonformuvausten derivaatan korkeampaan integroituvuuteen (Gehring, 1973). Tässä työssä tutkitaan eräitä dimension muuntumiseen liittyviä ongelmia kvasikonformikuvauksilla sekä euklidessa tasossa että n-avaruudessa. Erityistä huomiota kohdistetaan kvasiympyröitten ja kvasipallojen dimensioon, jotka ovat ympyröiden ja pallojen kuvajoukkoja kvasikonformikuvauksissa

Burkholder Integrals, Morrey\u27s Problem and Quasiconformal Mappings

Inspired by Morrey\u27s Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals Bp, p \u3e 2, are quasiconcave, when tested on deformations of identity f in Id+Coinifinty (omega) with Bp (Df(x)) \u3e 0 pointwise, or equivalently, deformations such that abs[Df]2 \u3c (p/(p-2))Jf. In particular, this holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible Lp-estimates for the gradient of a principal solution to the Beltrami equation fz = mu(z)fz , for any p in the critical interval 2 \u3c 1+1/ abs[mu f]infinity. Examples of local maxima lacking symmetry manifest the intricate nature of the problem

Disgust trumps lust:women’s disgust and attraction towards men is unaffected by sexual arousal

Mating is a double-edged sword. It can have great adaptive benefits, but also high costs, depending on the mate. Disgust is an avoidance reaction that serves the function of discouraging costly mating decisions, for example if the risk of pathogen transmission is high. It should, however, be temporarily inhibited in order to enable potentially adaptive mating. We therefore tested the hypothesis that sexual arousal inhibits disgust if a partner is attractive, but not if he is unattractive or shows signs of disease. In an online experiment, women rated their disgust towards anticipated behaviors with men depicted on photographs. Participants did so in a sexually aroused state and in a control state. The faces varied in attractiveness and the presence of disease cues (blemishes). We found that disease cues and attractiveness, but not sexual arousal, influenced disgust. The results suggest that women feel disgust at sexual contact with unattractive or diseased men independently of their sexual arousal