Non-minimality of corners in subriemannian geometry

Abstract

The regularity of length-minimizing curves on subriemannian manifolds is an open problem, of which little is known outside of some specific cases. One of the only general results is a theorem of Sussmann stating that all length-minimizers are analytic on an open dense subset of their domains. Our result allows one to exclude a class of extremals from the study of length-minimizing curves. That is, we prove that a curve with a corner-type singularity cannot be length-minimizing. The result is based on an explicit shortening procedure for corners in Carnot groups, from which the result for subriemannian manifolds follows by a desingularization and blow-up argument. This talk is based on joint work with Enrico Le Donne.