$h$-Principle for Curves with Prescribed Curvature

TL;DR

This paper establishes a $C^1$-dense h-principle for approximating immersed curves in $ ^n$ with prescribed curvature, enabling the construction of knots with arbitrary positive curvature in any isotopy class.

Contribution

It proves a $C^1$-dense h-principle for curves with prescribed curvature, extending previous results to more general curvature functions and applications to knot theory.

Findings

Any immersed $C^2$-curve can be approximated by curves with prescribed curvature greater than the original.

Existence of $C^2$-knots with arbitrary positive curvature in each isotopy class.

Generalization of McAtee's result to variable positive curvature.

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.