Abstract.
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.
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Received: July 2006, Revision: October 2006, Accepted: November 2006
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Leonardi, G.P., Monti, R. End-Point Equations and Regularity of Sub-Riemannian Geodesics. GAFA Geom. funct. anal. 18, 552–582 (2008). https://doi.org/10.1007/s00039-008-0662-y
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DOI: https://doi.org/10.1007/s00039-008-0662-y