Sharp regularity of sub-Riemannian length-minimizing curves

TL;DR

This work investigates the regularity of length-minimizing horizontal curves in a class of sub-Riemannian structures on $\mathbb{R}^3$ defined by $X_1=\partial_{x_1}$ and $X_2=\partial_{x_2}+P(x)^2\partial_{x_3}$ with $P(x)=x_1^a-x_2^b$. The abnormal candidate geodesic $\gamma(t)=(t^b,t^a,0)$ is shown to fail length-minimality for a range of parameter pairs, establishing sharpness of prior results that yielded length-minimizing behavior in other regimes. The authors construct explicit shorter competitors by projecting to $\mathbb{R}^2$ and solving a constrained isoperimetric problem encoded via Stokes theory, balancing a weighted area against a third-coordinate error. These results clarify how the integers $a$ and $b$ govern minimality and regularity, and they point toward deeper structural questions about when $C^2$ regularity can be guaranteed or when truly non-smooth geodesics arise. The approach provides a concrete tool for analyzing abnormal curves and highlights the delicate interplay between analytic and geometric features in sub-Riemannian geometry.

Abstract

A longstanding open question in sub-Riemannian geometry is the smoothness of (the arc-length parameterization of) length-minimizing curves. In [6], this question is negative answered, with an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure. In this paper, we study a class of examples of sub-Riemannian structures that generalizes that presented in [6], and we prove that length-minimizing curves must be at least of class $C^2$ within these examples. In particular, we prove that Theorem 1.1 in [6] is sharp.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• Lemma 2.6
• proof
• Definition 2.7