New properties of length-extremals in free step-2 rank-4 Carnot groups
Annamaria Montanari, Daniele Morbidelli
TL;DR
This work analyzes the free step-2 Carnot group of rank 4 to derive an explicit expression for length-minimizing curves from the origin and an explicit equation for conjugate points. The key innovation is a factorization of the conjugate-point equation into a part tied to the conjectured cut locus $C_4$ and another capturing additional conjugate points, enabling connections to the Rizzi–Serres cut-locus picture and to finite cut-time results for a broad class of extremals. The authors exploit a change of basis to simplify the extremal expressions, develop a Hamiltonian framework, and perform a detailed conjugacy analysis via the differential of the endpoint map after mapping to a canonical parameter space. They also show that extremals lying in Carnot subgroups reduce to lower-rank analyses, and they provide upper bounds on the cut-time in both rational and irrational angular-parameter regimes, establishing finiteness of cut-time for nonrectilinear normal extremals. These results advance understanding of the cut locus structure in rank-4 free step-2 Carnot groups and tie in to the conjectured $C_4$-based description of the origin’s cut locus.
Abstract
In the free, step-2, rank-4 sub-Riemannian Carnot group, we give a clean expression for length-extremals, we provide an explicit equation for conjugate points, we relate it with the conjectured cut-locus of the origin. Finally, we give some upper estimates for the cut-time of extremals.