A two-step ideal sub-Riemannian structure with no measure contraction properties

Abstract

We construct a compact, equiregular, ideal sub-Riemannian manifold of step 2 such that, for every smooth positive measure, the MCP(K, N) fails for all K ∈ R and N ∈ (1, ∞). This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in [Ann. Inst. Fourier, 2020] cannot be replaced by smoothness. Moreover, our structure is ideal, i.e. it admits no non-trivial abnormal minimizing geodesics. Although failures of the measure contraction property for ideal structures were recently obtained in the higher-step setting, our construction shows that the phenomenon can already occur on compact, equiregular, ideal structures of step 2. The proof exploits a differential consequence of the measure contraction property, namely a uniform upper bound for the sub-Laplacian of the squared distance near its base point, and constructs a structure for which this quantity is unbounded.

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BibTeX RIS

BibTeX

@misc{rizzi2026twostep,
  title = {{A two-step ideal sub-Riemannian structure with no measure contraction properties}},
  author = {Luca Rizzi and Ye Zhang},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29698},
  eprint = {2609.29698},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05753931v1},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - A two-step ideal sub-Riemannian structure with no measure contraction properties
AU  - Luca Rizzi
AU  - Ye Zhang
PY  - 2026
DA  - 2026/09/01
PB  - HAL
DO  - 10.48550/arxiv.2609.29698
UR  - https://hal.science/hal-05753931v1
AB  - We construct a compact, equiregular, ideal sub-Riemannian manifold of step 2 such that, for every smooth positive measure, the MCP(K, N) fails for all K ∈ R and N ∈ (1, ∞). This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in [Ann. Inst. Fourier, 2020] cannot be replaced by smoothness. Moreover, our structure is ideal, i.e. it admits no non-trivial abnormal minimizing geodesics. Although failures of the measure contraction property for ideal structures were recently obtained in the higher-step setting, our construction shows that the phenomenon can already occur on compact, equiregular, ideal structures of step 2. The proof exploits a differential consequence of the measure contraction property, namely a uniform upper bound for the sub-Laplacian of the squared distance near its base point, and constructs a structure for which this quantity is unbounded.
ER  -

Abstract

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