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 x1, x2. 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.

End-point equations and regularity of sub-Riemannian geodesics / Leonardi, Gian Paolo; Roberto, Monti. - In: GEOMETRIC AND FUNCTIONAL ANALYSIS. - ISSN 1016-443X. - STAMPA. - 18:2(2008), pp. 552-582. [10.1007/s00039-008-0662-y]

End-point equations and regularity of sub-Riemannian geodesics

LEONARDI, Gian Paolo;
2008-01-01

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 x1, x2. 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.
2008
2
Leonardi, Gian Paolo; Roberto, Monti
End-point equations and regularity of sub-Riemannian geodesics / Leonardi, Gian Paolo; Roberto, Monti. - In: GEOMETRIC AND FUNCTIONAL ANALYSIS. - ISSN 1016-443X. - STAMPA. - 18:2(2008), pp. 552-582. [10.1007/s00039-008-0662-y]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/229170
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