End-point equations and regularity of sub-Riemannian geodesics

For a large class of equiregular sub-Riemannian manifolds, we showthat length minimizing curves have no corner-like singularities. Our first resultis the reduction of the problem to the homogeneous, rank 2 case, by means of anilpotent approximation. We also identify a suitable condition on the tangent Liealgebra implying existence of a horizontal basis of vector fields whose coefficientsdepend only on the first two coordinates x1, x2. Then, we cut the corner andlift the new curve to a horizontal one, obtaining a decrease of length as well as aperturbation of the end-point. In order to restore the end-point at a lower costof length, we introduce a new iterative construction, which represents the maincontribution of the paper. We also apply our results to some examples.