On the regularity of abnormal minimizers for rank $2$ sub-Riemannian structures

TL;DR

This paper proves the $C^{1}$ regularity of abnormal length-minimizers in rank 2 sub-Riemannian structures, showing that all such minimizers are $C^{1}$ for structures of step up to 4, advancing understanding of their smoothness.

Contribution

It establishes $C^{1}$ regularity for abnormal minimizers in rank 2 sub-Riemannian structures, extending to all structures of step up to 4, which was previously unknown.

Findings

All abnormal length-minimizers in rank 2 sub-Riemannian structures of step up to 4 are $C^{1}$.

The $C^{1}$ regularity holds for a broad class of abnormal minimizers.

The result improves understanding of the regularity properties of sub-Riemannian geodesics.

Abstract

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.