On Regularity of Abnormal Subriemannian Geodesics

TL;DR

This paper investigates the regularity and existence of abnormal geodesics in subriemannian manifolds, establishing smoothness results and non-existence of certain minimizers in specific Carnot groups, extending previous findings.

Contribution

It proves smoothness of abnormal minimizers in step 3 subriemannian manifolds with nilpotent basis and shows no strictly abnormal minimizers exist in rank 2, step 4 Carnot groups, also generalizing regularity results.

Findings

Abnormal minimizers are smooth in step 3 manifolds with nilpotent basis.

No strictly abnormal minimizers in rank 2, step 4 Carnot groups.

Abnormal minimizers lack corner singularities in manifolds of step less than 7.

Abstract

We prove the smoothness of abnormal minimizers of subriemannian manifolds of step 3 with a nilpotent basis. We prove that rank 2 Carnot groups of step 4 admit no strictly abnormal minimizers. For any subriemannian manifolds of step less than 7, we show all abnormal minimizers have no corner type singularities, which partly generalize the main result of Leonardi-Monti.