| A Cornucopia Of Four-Dimensional Abnormal Subriemannian Minimizers (1996) | |||||||||||||
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| . 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 extr... | |||||||||||||
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