$h$-Principles for smooth curves and knots with prescribed curvature

Abstract

We show that smooth curves with prescribed curvature satisfy a $C^1$-dense $h$-principle in the space of immersed curves in Euclidean space. More precisely, every $C^{\alpha \ge 2}$ curve with nonvanishing curvature in $R^{n\ge 3}$ can be $C^1$-approximated by $C^{\alpha }$ curves of any larger curvature, prescribed as a function of arclength. It follows that there exist $C^{\infty }$ knots of prescribed curvature in every isotopy class of closed curves embedded in $R^3$.

7 pages, 1 figure