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Theses

Homéomorphismes quasiconformes extrémaux et différentielles quadratiques en géométrie CR sphérique

Abstract : In this thesis, we are interested in the idea of Teichmüller homeomorphisms in the setting of 3-dimensional spherical CR geometry. We then consider two approaches of it. The first one is about quasiconformal mappings which have extremal properties. In this direction, we construct explicitly and prove uniqueness (up to composition with a rotation around the vertical axis) of a minimizer of a mean distortion between cylinders in the Heisenberg group. After that, we extend those results to lifts, by a natural projection in the upper half-plane, of quadrilaterals. We also prove a general result about lifting quasiconformal from the upper half-plane to the Heisenberg group. The second approach is the one of quasiconformal mappings which dilate hori- zontal trajectories of CR quadratic differentials. For that, we need to define what are CR quadratic differentials. We manage to do it by considering a decomposition of Rumin complex on spherical CR manifolds. Finally, we give several examples of quasiconformal mappings which dilate (horizontal and/or vertical) trajectories of CR quadratic differentials. In particular, we will see that, under certain conditions, there is at most a two-parameters family of quasiconformal mappings which dilate horizontal trajectories of quadratic differentials.
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  • HAL Id : tel-02900783, version 1

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Robin Timsit. Homéomorphismes quasiconformes extrémaux et différentielles quadratiques en géométrie CR sphérique. Géométrie différentielle [math.DG]. Sorbonne Université, 2018. Français. ⟨NNT : 2018SORUS602⟩. ⟨tel-02900783⟩

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