End-Point Equations and Regularity of Sub-Riemannian Geodesics

For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x ₁, x ₂. Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples. Received: July 2006, Revision: October 2006, Accepted: November 2006