A $C^m$ Whitney Extension Theorem for Horizontal Curves in Free Step $2$ Carnot Groups

TL;DR

This paper proves a $C^m$ Whitney extension theorem for horizontal curves in free step 2 Carnot groups, extending prior results in the Heisenberg group and introducing new techniques for handling vertical areas.

Contribution

It generalizes Whitney extension results to all free step 2 Carnot groups with multiple generators, using novel methods for analyzing jets and vertical area corrections.

Findings

Established a $C^m$ Whitney extension theorem for free step 2 Carnot groups.

Developed a quantitative linear dependence structure on jets.

Introduced a discretization process to correct vertical areas.

Abstract

We establish a $C^m$ Whitney extension theorem for horizontal curves in free step~$2$ Carnot groups $\mathbb{G}_r$ for an arbitrary number of generators $r \geq 2$. This extends existing results in the Heisenberg group. New techniques include a quantitative linear dependence structure on jets on the horizontal layer, a discretization process to correct vertical areas, and studying the effect of unrelated vertical areas.