h-Principles for curves and knots of constant curvature

Abstract

We prove that ${\mathcal{C}}^\infty$ curves of constant curvature satisfy, in the sense of Gromov, the relative ${\mathcal{C}}^1$-dense h-principle in the space of immersed curves in Euclidean space R ^{n ≥ 3}. In particular, in the isotopy class of any given ${\mathcal{C}}^1$ knot f there exists a ${\mathcal{C}}^\infty$ knot f͂ of constant curvature which is ${\mathcal{C}}^1$-close to f. More importantly, we show that if f is ${\mathcal{C}}^2$, then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull.