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Not all sub-Riemannian minimizing geodesics are smooth

Yacine Chitour, Frédéric Jean, Roberto Monti, Ludovic Rifford, Ludovic Sacchelli, Mario Sigalotti, Alessandro Socionovo

TL;DR

The paper answers a central question in sub-Riemannian geometry by constructing a real-analytic, polynomial sub-Riemannian structure in $\mathbb{R}^3$ that admits a length-minimizing horizontal curve which is not $C^\infty$; specifically, for odd $m\ge5$ the singular minimizer along the Martinet surface yields an arc-length reparametrization of class $C^{\bar m-1/2}$ but not $C^{\bar m+1/2}$, with the minimal example being $C^2$ but not $C^3$ when $m=5$. The proof strategy blends projection to the $(x_1,x_2)$-plane, the normal-extremal system with a Lagrange multiplier, and a delicate calculus-of-variations analysis constrained by $P$-sublevel sets. A central technical contribution is proving that the projected curve $\omega$ must acquire a single loop with precisely controlled geometry, and that any would-be smoother minimizer would contradict Stokes’ theorem, the isoperimetric inequality, or Gauss–Bonnet considerations. This result settles the question of universal smoothness for sub-Riemannian minimizing geodesics in the negative and sharpens understanding of singular geodesic regularity in real-analytic settings.

Abstract

A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.

Not all sub-Riemannian minimizing geodesics are smooth

TL;DR

The paper answers a central question in sub-Riemannian geometry by constructing a real-analytic, polynomial sub-Riemannian structure in that admits a length-minimizing horizontal curve which is not ; specifically, for odd the singular minimizer along the Martinet surface yields an arc-length reparametrization of class but not , with the minimal example being but not when . The proof strategy blends projection to the -plane, the normal-extremal system with a Lagrange multiplier, and a delicate calculus-of-variations analysis constrained by -sublevel sets. A central technical contribution is proving that the projected curve must acquire a single loop with precisely controlled geometry, and that any would-be smoother minimizer would contradict Stokes’ theorem, the isoperimetric inequality, or Gauss–Bonnet considerations. This result settles the question of universal smoothness for sub-Riemannian minimizing geodesics in the negative and sharpens understanding of singular geodesic regularity in real-analytic settings.

Abstract

A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a but not length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.

Paper Structure

This paper contains 11 sections, 17 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['MainTHM']}
  3. Proof of Proposition \ref{['PROPMain']}
  4. Reminders on Stokes' theorem and the isoperimetric inequality
  5. Curves with curvature of constant sign in the plane
  6. Preliminary observations on the curve $\omega$
  7. Some obstructions to the minimality of $\omega$
  8. Intersections of $\omega$ with $\{P=0\}$ and loops
  9. A closer look at the first loop of $\omega$
  10. End of proof of Proposition \ref{['PROPMain']}
  11. Proof of Proposition \ref{['PROPsublevelsets']}

Key Result

Theorem 1.1

For every odd integer $m\geq 5$ and for any sufficiently small $\epsilon>0$, the curve $\bar{\gamma}|_{ [0,\epsilon]}$ is the unique horizontal path minimizing the distance between $\bar{\gamma}(0)$ and $\bar{\gamma}(\epsilon)$ with respect to $(\Delta,g)$. Furthermore, its arc length reparametrizat

Figures (2)

  • Figure 1: A drawing of $\bar{\omega}$ and $\omega$
  • Figure 2: The curve $\nu_{\epsilon}^{\rho}$ in black

Theorems & Definitions (30)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 20 more