The regularity of energy-minimizing curves on subriemannian manifolds is an open problem, of which quite little is known outside of some specific cases. In the Riemannian setting, all minimizers are solutions to a geodesic equation, which implies that all minimizers are smooth. The main difference in the subriemannian case is the existence of so called abnormal extremals, which do not satisfy any geodesic equation, but may nonetheless be minimizers.
In this talk, I will outline a method used to prove that every tangent of a minimizer is still a minimizer when projected onto some structure of lower nilpotency step. In particular, we can prove that a piecewise C^1 minimizer is necessarily everywhere C^1, that is, that curves with 'corners' cannot be length-minimizing. This talk is based on joint work with Eero Hakavuori.