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

It is a longstanding problem in sub-Riemannian geometry whether length-minimizing curves are smooth. It is known that normal extremals are smooth, but the case of abnormal minimizers is still open. We present an improvement of the existing partial results that guarantees the $C^1$ regularity for a class of abnormal length-minimizers in rank 2 sub-Riemannian structures. As a consequence of such a result, all length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class $C^1$. This is a joint work with Davide Barilari, Yacine Chitour, Frédéric Jean and Mario Sigalotti.