A cornucopia of four-dimensional abnormal sub-Riemannian minimizers

Abstract

We study in detail the local optimality of abnormal sub-Riemannian extremals for a completely arbitrary sub-Riemannian structure on a four-dimensional manifold, associated to a two-dimensional bracket-generating regular distribution. Using a technique introduced in earlier work with W. Liu, we show that large collections of simple (i.e. without double points) nondegenerate extremals exist, and are always uniquely locally optimal. In particular, we prove that the simple abnormal extremals parametrized by arc-length foliate the space (i.e. through every point there passes exactly one of them) and they are all local minimizers. Under an extra nondegeneracy assumption, these abnormal extremals are strictly abnormal (i.e. are not normal). (In the forthcoming paper [6] with W. Liu we show that in higher dimensions there are large families of “nondegenerate abnormal extremals” that are local minimizers as well. In dimension 3, for a regular distribution there are no nontrivial abnormal extremals at all, but if the distribution is not regular then, generically, there are two-dimensional surfaces that are foliated by abnormal extremals, all of which turn out to be local minimizers.) This adds up to a picture which is rather different from the one that appeared to emerge from previous work by R. Montgomery and I. Kupka, in which an example of an abnormal extremal for a nonregular distribution in ℝ3 was studied and shown to be locally optimal with great effort, by means of a very long and laborious argument, and then this example was used to produce a similar one for a regular distribution in ℝ4. All this may have given the impression that abnormal extremals are hard to find, and that proving them to be minimizers is an arduous task that can only be accomplished in some very exceptional cases. Our results show that abnormal extremals exist aplenty, that most of them are local minimizers, and that in some widely studied cases, such as regular distributions on ℝ4, this is in fact true for all of them.