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Theses

Intégration convexe effective

Abstract : The aim of this thesis is to propose an effective version of Convex Integration Theory. This theory, developed by M. Gromov in the 70’s, allows to solve differential relations, i.e. partial differential equalities / inequalities. In this thesis, we introduce a formula called Corrugation Process. The key formula of the Convex Integration Theory can be substituted by this new formula. The expression of the Corrugation Process is interesting for the relations characterized in this thesis: the Kuiper relations. We show that this kind of relation appears in differential geometry, for example for immersions, for isometric immersions and for totally real maps. In particular, the results obtained in this thesis allow to build directly a new immersion of RP^2. The Corrugation Process and the notion of Kuiper relation offer a natural framework to study potential self-similarity properties. Such properties were already observed for the C^1-isometric embedding of a flat torus and of a reduced sphere built by the Hevea team. Precisely, in this thesis, we show a self-similarity property for some C^1-isometric totally real embeddings
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Mélanie Theillière. Intégration convexe effective. Géométrie différentielle [math.DG]. Université de Lyon, 2019. Français. ⟨NNT : 2019LYSE1342⟩. ⟨tel-02524194⟩

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