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arXiv · 2107.04554

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

Scott Zimmerman
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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 $C^{m,\sqrtω}$ horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman.

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Scott Zimmerman. 2022-01-20. Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group. https://arxiv.org/abs/2107.04554

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