Non-minimality of spirals in sub-Riemannian manifolds

Abstract

We show that in analytic sub-Riemannian manifolds of rank 2 satisfying a commutativity condition spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve.

Higher order Goh conditions for singular extremals of corank 1

“…A recent approach to the regularity problem of length-minimizing curves is based on the analysis of specific singularities such as corners, spiral-like curves or curves with no straight tangent line. This approach does not use general open mapping theorems but it rather relies on the ad hoc construction of shorter competitors, see [4,10,11,16,20,21,22].…”