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Unrectifiable surfaces with intrinsic regularity in Carnot groups

  • Valentino Magnani
  • Jeremy Mirmina
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Abstract

In any noncommutative Carnot group, for every 0≤ α<1, we show the existence of a C1,α H surface that is unrectifiable with respect to the Euclidean distance. Precisely, the surface has Euclidean Hausdorff dimension higher than its topological dimension. We show that in step-2 Carnot groups the construction gives hypersurfaces of maximal Euclidean Hausdorff dimension, equal to q−(1+α)/2, where q is the topological dimension of the group. Here the range of unrectifiability is sharp, since for α=1 the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the (q−1/2)-dimensional Euclidean Hausdorff measure of every C1 H hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.

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Cite

BibTeXRIS
BibTeX
@misc{magnani2026unrectifiable,
  title = {{Unrectifiable surfaces with intrinsic regularity in Carnot groups}},
  author = {Valentino Magnani and Jeremy Mirmina},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15851},
  eprint = {2610.15851},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15851},
}
RIS
TY  - UNPB
TI  - Unrectifiable surfaces with intrinsic regularity in Carnot groups
AU  - Valentino Magnani
AU  - Jeremy Mirmina
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15851
UR  - https://arxiv.org/abs/2610.15851
AB  - In any noncommutative Carnot group, for every 0≤ α<1, we show the existence of a C^1,α_ H surface that is unrectifiable with respect to the Euclidean distance. Precisely, the surface has Euclidean Hausdorff dimension higher than its topological dimension. We show that in step-2 Carnot groups the construction gives hypersurfaces of maximal Euclidean Hausdorff dimension, equal to q−(1+α)/2, where q is the topological dimension of the group. Here the range of unrectifiability is sharp, since for α=1 the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the (q−1/2)-dimensional Euclidean Hausdorff measure of every C^1_ H hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.
ER  -