On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures

Abstract

We prove the C 1 regularity for a class of abnormal length-minimizers in rank 2 sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class C 1 . arXiv:1804.00971v2 [math.OC] 4 Dec 2018If the sub-Riemannian manifold has rank 2 and step at most 4, the assumption in Theorem 1 is trivially satisfied by every abnormal minimizer γ and we immediately obtain the following corollary.Corollary 2. Assume that the sub-Riemannian structure has rank 2 and step at most 4. Then all length-minimizers are of class C 1 .It is legitimate to ask whether the C 1 regularity in the Open Problem can be further improved. Indeed, the argument behind our proof permits to obtain C ∞ regularity of length-minimizers under an additional nilpotency condition on the Lie algebra generated by horizontal vector fields.Proposition 3. Assume that D is generated by two vector fields X 1 , X 2 such that the Lie algebra Lie{X 1 , X 2 } is nilpotent of step at most 4. Then for every sub-Riemannian structure (D, g) on M , the corresponding length-minimizers are of class C ∞ .The above proposition applies in particular to Carnot groups of rank 2 and step at most 4. In this case we recover the results obtained in [LM08, Example 4.6].The strategy of proof of Theorem 1 is to show that, at points where they are not of class C 1 , length-minimizers can admit only corner-like singularities. This is done by a careful asymptotic analysis of the differential equations satisfied by the abnormal lift, which exploits their Hamiltonian structure. We can then conclude thanks to the following result. Theorem 4 ([HL16]). Let M be a sub-Riemannian manifold. Let T > 0 and let γ : [−T, T ] → M be a horizontal curve parametrized by arclength. Assume that, in local

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