Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

TL;DR

This paper investigates rectifiability of smooth hypersurfaces in Carnot groups, showing that certain notions of rectifiability differ and providing examples of hypersurfaces with complex tangent structures, contrasting with the Heisenberg group case.

Contribution

It constructs a smooth hypersurface in a Carnot group with uncountably many non-isomorphic tangent groups, demonstrating differences between rectifiability notions and their implications.

Findings

Existence of a hypersurface with uncountably many tangent groups in a Carnot group.

Such hypersurface cannot be Lipschitz parametrized by countably many maps.

All smooth hypersurfaces in Heisenberg groups are countably rectifiable.

Abstract

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a $C^{\infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $\mathcal{H}^{12}$. As a consequence, we show that for every Carnot group of Hausdorff dimension 12, any Lipschitz map defined on a subset of it with values in $S$ has $\mathcal{H}^{12}$-null image. In particular, we deduce that this smooth hypersurface cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension $12$. As main consequence we have that a notion of rectifiability proposed by S.Pauls is not equivalent to one proposed by B.Franchi, R.Serapioni and F.Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset $U$ of a homogeneous subgroup of Hausdorff dimension $12$ of a Carnot group, every bi-Lipschitz map $f:U\to S$ satisfies $\mathcal{H}^{12}(f(U))=0$. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all $C^{\infty}$-hypersurfaces in $\mathbb H^n$ with $n\geq 2$ are countably $\mathbb{H}^{n-1}\times\mathbb R$-rectifiabile according to Pauls' definition, even with bi-Lipschitz maps.