Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mohammad Ghomi, Matteo Raffaelli

TL;DR

The paper proves a $C^1$-dense $h$-principle for smooth curves with prescribed curvature in $\mathbb{R}^n$, showing that any $C^{\alpha\ge2}$ immersed curve with positive curvature can be $C^1$-approximated by curves of larger, prescribed curvature while preserving unit-speed and tangency at prescribed points. The core strategy converts the global problem into a local spherical problem for the tantrix, using convex integration, a Kalman-type barycentric construction, and degree theory to control averages. The authors provide a constructive path from unit-speed curves to prescribed-curvature curves, and deduce the existence of $C^\infty$ knots within each isotopy class with prescribed curvature, thereby strengthening prior results and enabling finer control over speed and position. This framework yields new isotopy-type consequences for knots and broadens the applicability of convex-integration techniques to geometric curve problems with curvature constraints.

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 $C^{α\geq 2}$ curve with nonvanishing curvature in $R^{n\geq 3}$ can be $C^1$-approximated by $C^α$ 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 $R^3$.

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

TL;DR

The paper proves a -dense -principle for smooth curves with prescribed curvature in , showing that any immersed curve with positive curvature can be -approximated by curves of larger, prescribed curvature while preserving unit-speed and tangency at prescribed points. The core strategy converts the global problem into a local spherical problem for the tantrix, using convex integration, a Kalman-type barycentric construction, and degree theory to control averages. The authors provide a constructive path from unit-speed curves to prescribed-curvature curves, and deduce the existence of knots within each isotopy class with prescribed curvature, thereby strengthening prior results and enabling finer control over speed and position. This framework yields new isotopy-type consequences for knots and broadens the applicability of convex-integration techniques to geometric curve problems with curvature constraints.

Abstract

We show that smooth curves with prescribed curvature satisfy a -dense -principle in the space of immersed curves in Euclidean space. More precisely, every curve with nonvanishing curvature in can be -approximated by curves of any larger curvature, prescribed as a function of arclength. It follows that there exist knots of prescribed curvature in every isotopy class of closed curves embedded in .

Paper Structure

This paper contains 6 sections, 6 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unit speed reparametrization
  4. Average and center of mass
  5. Reduction to Spherical Curves
  6. Proof of Theorem \ref{['TH:main']}

Key Result

Theorem 1.1

Let $f\in\mathop{\mathrm{Imm}}\nolimits^{\alpha\geqslant 2}(\Gamma,\mathbf{R}^{n\geqslant 3})$ be a curve with curvature $\kappa>0$, and $\widetilde{\kappa}\colon\Gamma\to\mathbf{R}$ be a $\mathcal{C}^{\alpha-2}$ function with $\widetilde{\kappa}>\kappa$. Then, for any $\varepsilon>0$, there exists

Figures (1)

  • Figure 1:

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1: ghomi-raffaelli2024b
  • Lemma 4.2
  • proof
  • proof : Proof of Proposition \ref{['PROP:spherical']}