Sharp regularity of sub-Riemannian length-minimizing curves

TL;DR

This paper investigates the smoothness of length-minimizing curves in sub-Riemannian geometry, proving they are at least twice differentiable in certain generalized structures, thus confirming the sharpness of previous results.

Contribution

It generalizes prior examples and establishes that length-minimizers are at least C^2, confirming the optimality of earlier smoothness bounds in sub-Riemannian geometry.

Findings

Length-minimizers are at least C^2 in the studied structures.

Theorem 1.1 in [6] is shown to be sharp.

Provides a broader class of examples with smoothness properties.

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.