On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups

TL;DR

This paper investigates the conditions under which the subRiemannian distance in certain Carnot groups exhibits semiconcavity, revealing that it depends on the presence of abnormal minimizing curves and the group's step.

Contribution

It provides explicit estimates showing when subRiemannian distance is semiconcave in step-two Carnot groups and demonstrates failure of semiconcavity in the Engel group.

Findings

Semiconcavity holds in step-two groups without abnormal curves

Semiconcavity fails in the Engel group even horizontally

Explicit estimates characterize semiconcavity conditions

Abstract

We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizing curves. Moreover, we prove that local semiconcavity fails to hold in the step-3 Engel group, even in the weaker "horizontal" sense.