An upper bound for the length of a Traveling Salesman path in the Heisenberg group

Abstract

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$β$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.

19 pages. No figures; V2 several (inconsequential) errors corrected. V3 minor changes. Accepted to Revista Matemática Iberoamericana