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.

“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 façade 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?”

G.K. Chesterton, “The Three Tools Of Death,” in The Innocence of Father Brown, The Father Brown Omnibus, Dodd, Mead & Co., New York (1983), p. 117.

This work was done while the author was a visitor at Institute for Mathematics and its Applications (I.M.A.), University of Minnesota, Minneapolis, Minnesota 55455, U.S.A. A preprint of this paper, entitled “A cornucopia of abnormal sub-Riemannian minimizers, Part I: the four-dimensional case,” is # 1073 of the I.M.A. Preprint series, December 1992.

Supported in part by the National Science Foundation under Grant DMS92-02554.