Abnormal extremals and optimality in sub-Riemannian manifolds

One of the most important questions in sub-Riemannian geometry is whether optimal abnormal extremals can exist. This question remained open for several years, and was settled by Montgomery (1993), who exhibited a counterexample. But Montgomery´s geometric optimality proof depends heavily on special properties of his example and still leaves open the question whether optimal abnormal extremals are an exceptional phenomenon or a common occurrence. In this paper we study length minimizing arcs in sub-Riemannian manifolds (M, E, G) where the metric G is defined on a rank-two bracket-generating distribution E. We present an analytic technique for proving local optimality of a large class of abnormal extremals that we call “regular”. If E satisfies a mild additional restriction-valid in particular for all regular 2-dimensional distributions and for generic 2-dimensional distributions-then regular abnormal extremals are “typical”, so our result implies that the abnormal minimizers are ubiquitous rather than exceptional