The regularity problem for sub-Riemannian geodesics

The first section of this thesis aims to illustrate the regularity problem for geodesics in sub-Riemannian manifolds, which can be intended as metric spaces when endowed with the Carnot-Caratheodory if Hörmander's condition holds for the horizontal sub-bundle. We are able to impose some first-order necessary conditions for length-minimizing curves, which then may classify the extremals, i.e. the class of curves that happen to satisfy these conditions, as normal or abnormal (or both). While normal extremals are automatically C∞ smooth, as of April 2024 there are only partial results regarding the regularity of (strictly) abnormal extremals. It is a well-known fact that the tangent metric space (in the Gromov-Hausdorff sense) to a C-C space at a “generic” point is a Carnot group, i.e. a stratified Lie group endowed with a C-C structure. Therefore, the second section of this thesis focuses on Carnot Groups and some of their basic properties, together with the introduction of tools to study geodesics in Carnot Groups. The last section of this work presents a short argument by Eero Hakavuori and Enrico le Donne, which shows the non-minimality of corner-type singularities in Carnot Groups (hence in sub-Riemannian geometry in general). The proof is by induction on the step "s" of the group, and it revolves around isometrically projecting the Carnot group into a quotient group of step s-1. The inductive hypothesis guarantees the existence of a length minimizer in the quotient group, which can then be lifted back to the original one to obtain a curve shorter than the initial corner-type curve, but with an error in the endpoint of order s. However, this error can be corrected by placing a system of curves along the original one in an efficient enough way, thanks to the crucial fact that the space is a nilpotent and stratified group.