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∞-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 H12. As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in S, has negligible image with respect to the Hausdorff measure H12. In particular, we deduce that S 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→S satisfies H12(f(U))=0. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all C∞-hypersurfaces in Hn with n≥2 are countably Hn−1×R-rectifiable according to Pauls’ definition, even with bi-Lipschitz maps.
Highlights
State of the art: Measure-theoretic notions of rectifiability in Carnot groups have been deeply studied in the last 20 years
They proved that the reduced boundary of a set of finite perimeter is codimension-one CH1 -rectifiable showing that this could be a good notion of rectifiability at least for Heisenberg groups
In [22] the authors provide an implicit function theorem for CH1 -functions in the setting of Carnot groups, showing that a CH1 hypersurface is locally the boundary of a set of finite perimeter
Summary
State of the art: Measure-theoretic notions of rectifiability in Carnot groups have been deeply studied in the last 20 years. Still in [23], the authors proposed, in the setting of Heisenberg groups and for any dimension k, a definition of rectifiability a priori more general than k-dimensional CH1 -rectifiability, by using coverings with graphs of intrinsic Lipschitz functions. In this reference it is showed that a CH1 -hypersurface is locally the graph of a uniformly intrinsic differentiable function that solves a Burger-type equation Generalizations of this result are contained in [3] in the setting of CH1 -submanifolds in Hn, in [14] for CH1 -hypersurfaces in Carnot groups of step 2 and in [13] for CH1 -submanifolds in Carnot groups of step 2. Merlo for having shared with us a manuscript of [43]
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