In this paper, the authors prove that certain curves with corners are not length-minimizing in a class of sub-Riemannian manifolds. More precisely, consider an \(n\)-dimensional smooth manifold \(M\) and a bracket-generating smooth subbundle \({\mathcal D}\subset TM\) of rank \(m\geq 2\) generated (locally) by a family of smooth vectors fields \(X_1,\ldots,X_m\). Let \(\mathcal D_i(x)\), \(x\in M\), be the subspace generated by \(i\) iterated commutators of \(X_j\) and \(\mathcal L_i(x)=\mathcal D_1(x)+\ldots+\mathcal D_i(x)\). Assuming that the condition \[ \mathcal L_i(x)\neq \mathcal L_{i-1}(x)\Rightarrow \mathcal L_{i+1}(x)=\mathcal L_{i}(x), \quad\text{for all } i\geq 2, \] is satisfied at some point \(x\in M\), it is proved in Theorem 1.1 that horizontal piecewise smooth curves with corners at \(x\) are not length-minimizing.
The proof consists of a blow-up argument reducing the problem to the case of a piecewise linear curve in \(\mathbb R^n\) with a two-dimensional distribution, and an inductive argument on the dimension \(n\). The use of the Nagel-Stein-Wainger estimate (2.16) is crucial in the argument, see Theorem 4 in [A. Nagel et al., Acta Math. 155, 103–147 (1985; Zbl 0578.32044)].
As an application of Theorem 1.1, the authors show that in the sub-Riemannian manifold \((\mathbb R^4,\Delta,g)\), where \(\Delta\) is the \(2\)-dimensional distribution generated by \[ X_1={\partial\over \partial x_1}+2x_2 {\partial\over\partial x_3}+x_3^2{\partial\over\partial x_4},\qquad X_2={\partial\over\partial x_2}-2x_1{\partial\over\partial x_3}, \] and \(g\) is any smooth metric on \(\Delta\), all length-minimizing curves are smooth.
Related results can be found in [the authors, Geom. Funct. Anal. 23, No. 4, 1371–1401 (2013; Zbl 1278.53035)] and [the second and fourth author, Geom. Funct. Anal. 18, No. 2, 552–582 (2008; Zbl 1189.53033)].