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Convex integration theory without integration

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Abstract

We replace the usual Convex Integration formula by a Corrugation Process and introduce the notion of Kuiper differential relations. This notion provides a natural framework for the construction of solutions with self-similarity properties. We consider the case of the totally real relation, we prove that it is Kuiper and we state a totally real isometric embedding theorem. We then show that the totally real isometric embeddings obtained by the Corrugation Process exhibit a self-similarity property. Kuiper relations also enable a uniform expression of the Corrugation Process that no longer involves integrals. We apply the Corrugation Process to build a new explicit immersion of \({\mathbb {R}}P^2\) inside \({\mathbb {R}}^3\).

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Acknowledgements

The results presented here are a part of my PhD Thesis. I wish to express my gratitude towards the whole team of the Hevea project and specially to my advisors Vincent Borrelli and Boris Thibert for their help and support. I greatly thank Patrick Massot for his interest towards this work and many valuable comments that help me to improve the content of this text. I am grateful to the anonymous referee for their constructive and insightful comments.

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  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622, Villeurbanne Cedex, France

    Mélanie Theillière

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Correspondence to Mélanie Theillière.

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Theillière, M. Convex integration theory without integration. Math. Z. 300, 2737–2770 (2022). https://doi.org/10.1007/s00209-021-02785-9

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