A metric tangent approach to regularity of geodesics in sub-Riemannian manifolds


The regularity of length-minimizing curves on sub-Riemannian manifolds is an open problem, of which quite little is known compared to the Riemannian setting, where classically all minimizers are smooth. The main issue in the sub-Riemannian case is the existence of abnormal minimizers which do not satisfy any geodesic equation. In this talk I will give an overview of the problem and describe a series of results obtained by studying the tangent cone of a minimizing curve. The benefit of the approach is that it allows simplifying the problem to studying minimizers in sub-Riemannian Carnot groups. The group structure is then used for a constructive linear algebraic method to shorten curves, which can prove the non-minimality of some class of curves including for instance curves with corner type singularities. This talk is based on joint work with Enrico Le Donne.

Grenoble, France
Eero Hakavuori
Postdoc researcher