A Cornucopia of Four-dimensional Abnormal Subriemannian Minimizers 1

\The skull seems broken as with some big weapon, but there's no weapon at all lying about, and the murderer would have found it awkward to carry it away, unless the weapon was to small to be noticed." \Perhaps the weapon was too big to be noticed," said the priest, with an odd little giggle. Gilder looked round at this wild remark, and rather sternly asked Brown what he meant. \Silly way of putting it, I know," said Father Brown apologetically. \Sounds like a fairy tale. But poor Armstrong was killed with a giant's club, a great green club, too big to be seen, and which we call the earth. He was broken against this green bank we are standing on." \How do you mean?" asked the detective quickly. Father Brown turned his moon face up to the narrow faa cade of the house and blinked hopelessly up. Following his eyes, they saw that right at the top of this otherwise blind back quarter of the building, an attic window stood open. \Don't you see," he explained, pointing a little awkwardly like a child, \he was thrown down from there?" ABSTRACT. We study in detail the local optimality of abnormal sub-Riemannian ex-tremals for a completely arbitrary sub-Riemannian structure on a four-dimensional man-ifold, 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 diierent from the one that appeared to emerge from previous work by R. Montgomery and I. Kupka, in which an example of an abnormal extremal …