Calculus of Variations and Geometric Measure Theory

E. Le Donne - R. Montgomery - A. Ottazzi - P. Pansu - D. Vittone

Sard Property for the endpoint map on some Carnot groups

created by ledonne on 12 Mar 2015
modified by vittone on 25 Jul 2017


Published Paper

Inserted: 12 mar 2015
Last Updated: 25 jul 2017

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Volume: 33
Number: 6
Pages: 1639--1666
Year: 2016


In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

Keywords: Carnot groups, sub-Riemannian geometry, Abnormal curves, Sard's property, endpoint map, polarized groups