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h-Principle for curves with prescribed curvature

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Abstract

We prove that every immersed \(C^2\)-curve \(\gamma \) in \(\mathbb R^n\), \(n\geqslant 3\) with curvature \(k_{\gamma }\) can be \(C^1\)-approximated by immersed \(C^2\)-curves having prescribed curvature \(k>k_{\gamma }\). The approximating curves satisfy a \(C^1\)-dense h-principle. As an application we obtain the existence of \(C^2\)-knots of arbitrary positive curvature in each isotopy class, which generalizes a similar result by McAtee for \(C^2\)-knots of constant curvature.

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Acknowledgments

This article is part of my Ph.D. thesis. I would like to warmly thank my advisor Norbert Hungerbühler for his guidance, support and interest in this work. Furthermore I would like to thank Anand Dessai, Felix Hensel, Berit Singer, Nicolas Weisskopf and Thomas Mettler for helpful comments.

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Correspondence to Micha Wasem.

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Wasem, M. h-Principle for curves with prescribed curvature. Geom Dedicata 184, 135–142 (2016). https://doi.org/10.1007/s10711-016-0161-5

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  • DOI: https://doi.org/10.1007/s10711-016-0161-5

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