Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group

Consider the sub-Riemannian Heisenberg group H. In this paper, we answer the following question: given a compact set K ⊆ R and a continuous map f : K → H, when is there a horizontal C curve F : R → H such that F |K = f? Whitney originally answered this question for real valued mappings [35], and Fefferman provided a complete answer for real valued functions defined on subsets of R [12]. We also prove a finiteness principle for C √ ω horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman [5].