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arXiv · 1510.01954

$h$-Principle for Curves with Prescribed Curvature

Micha Wasem
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Abstract

We prove that every immersed $C^2$-curve $γ$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_γ$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_γ$. 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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Micha Wasem. 2016-04-14. $h$-Principle for Curves with Prescribed Curvature. https://doi.org/10.1007/s10711-016-0161-5

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