Not all sub-Riemannian minimizing geodesics are smooth

TL;DR

This paper provides a counterexample showing that not all sub-Riemannian length minimizers are smooth, answering a longstanding open question in the field.

Contribution

The authors construct a $C^2$ but not $C^3$ sub-Riemannian minimizer, demonstrating that smoothness of minimizers does not always hold.

Findings

Existence of a $C^2$ length minimizer that is not $C^3$

Counterexample in a real-analytic sub-Riemannian structure

Negative answer to the smoothness of all sub-Riemannian minimizers

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.