$h$-Principle for Curves with Prescribed Curvature

Abstract

We prove that every immersed $C^2$-curve $\gamma$ in ${\mathbb{R}}^n$, $n⩾3$ 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.

arXiv commentFinal version, to appear in Geometriae Dedicata, 9 pages, 1 figure