1. The extension problem
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Problem 1.1.
Given an Engel structure $\mathcal{D}$ on an open neighbourhood $U$ of the 3-sphere $S^3\subset\mathbb{R}^4$, when does there exist an Engel structure $\hat{\mathcal{D}}$ on $D^4$ that extends $\mathcal{D}$, i.e. \[ \hat{\mathcal{D}}=\mathcal{D}\ \text{on}\ D^4\cap U \] -
Problem 1.2.
Does a parametric version of the question in Problem 1.1 hold? For example, if we have a 1-parametric family of germs of Engel structures $\mathcal{D}_\lambda$ on $S^3\subset\mathbb{R}^4$ for $0\le\lambda\le1$ such that $\mathcal{D}_0$ and $\mathcal{D}_1$ extend to the disc $D^4$, does there exist a compatible extension for all $\lambda$?-
Remark. We assume that the formal obstructions to extension vanish.
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Remark. During the workshop, Problem 1.1. was answered positively by A. del Pino and T. Vogel, but their solution is not obviously parametric. So Problem 1.2 remains open, and various participants are working on versions of it.
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Problem 1.3.
Do there exist two Engel structures that are formally homotopic but not homotopic?-
Remark. If the most general version of Problem 1.2 has a positive solution, one expects Problem 1.3 to have a negative answer, since in that case we would have complete Engel flexibility. It is also possible that both problems have this answer (positive for 1.1., negative for 1.3) only in a category of flexible or overtwisted structures.
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Cite this as: AimPL: Engel structures, available at http://aimpl.org/engelstr.