On the cut locus of free, step two Carnot groups

Abstract

In this note, we study the cut locus of the free, step two Carnot groups $\mathbb{G}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exhibiting sets of cut points $C_k \subset \mathbb{G}_k$ which, for $k \geq 4$, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension $Θ(k^2)$ and semi-algebraic sets of codimension $Θ(k)$, the sets $C_k$ are semi-algebraic and have codimension $2$, yielding the best possible lower bound valid for all $k$ on the size of the cut locus of $\mathbb{G}_k$. Furthermore, we study the relation of the cut locus with the so-called abnormal set. In the low dimensional cases, it is known that \[ \mathrm{Abn}_0(\mathbb{G}_k) = \overline{\mathrm{Cut}_0(\mathbb{G}_k)} \setminus \mathrm{Cut}_0(\mathbb{G}_k), \qquad k=2,3. \] For each $k \geq 4$, instead, we show that the cut locus always intersects the abnormal set, and there are plenty of abnormal geodesics with finite cut time. Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of $C_k$. The question whether $C_k$ coincides with the cut locus for $k\geq 4$ remains open.