On the cut locus of free, step two Carnot groups

TL;DR

This paper investigates the structure of the cut locus in free, step two Carnot groups, disproving previous conjectures, revealing new properties of the cut locus, and exploring its relation to abnormal geodesics and heat kernel asymptotics.

Contribution

It provides the first explicit description of the cut locus in free, step two Carnot groups for all dimensions, challenging prior conjectures and establishing new bounds and properties.

Findings

The cut locus sets $C_k$ are semi-algebraic with codimension 2 for all $k \\geq 4$.

The cut locus always intersects the abnormal set for $k \\geq 4$.

An explicit lower bound for small time heat kernel asymptotics at points of $C_k$.

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-Carath\'eodory 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 $\Theta(k^2)$ and semi-algebraic sets of codimension $\Theta(k)$, 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.