Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group

TL;DR

This paper extends Whitney's classical extension theorem to the setting of horizontal curves in the Heisenberg group, providing conditions for extending continuous maps to smooth horizontal curves and establishing a finiteness principle in this sub-Riemannian context.

Contribution

It generalizes Whitney's extension theorem and the finiteness principle to horizontal curves in the Heisenberg group, a key step in sub-Riemannian geometry.

Findings

Characterization of when a continuous map from a compact set to the Heisenberg group extends to a horizontal $C^m$ curve.

Proof of a finiteness principle for $C^{m, oot{ ext{omega}}{ ext{sqrt}}}$ horizontal curves.

Extension of classical Whitney and Fefferman results to the sub-Riemannian setting.

Abstract

Consider the sub-Riemannian Heisenberg group $\mathbb{H}$. In this paper, we answer the following question: given a compact set $K \subseteq \mathbb{R}$ and a continuous map $f:K \to \mathbb{H}$, when is there a horizontal $C^m$ curve $F:\mathbb{R} \to \mathbb{H}$ such that $F|_K = f$? Whitney originally answered this question for real valued mappings, and Fefferman provided a complete answer for real valued functions defined on subsets of $\mathbb{R}^n$. We also prove a finiteness principle for $C^{m,\sqrt{\omega}}$ horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman.