{
  "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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  "article_type": null,
  "arxiv": "2609.29698",
  "author_total": 2,
  "authors": [
    {
      "name": "Luca Rizzi"
    },
    {
      "name": "Ye Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0q5e1rhk5v6w6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29698",
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  "id": "0q5e1rhk5v6w6",
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  "landing_url": "https://hal.science/hal-05753931v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
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  "pdf_url": "https://hal.science/hal-05753931/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0q5e1rhk5v6w6.ris",
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      "first_seen": "2026-10-30T12:17:36Z",
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      "url": "https://hal.science/hal-05753931v1"
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      "native_id": "2609.29698",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29698"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A two-step ideal sub-Riemannian structure with no measure contraction properties",
  "title_text": "A two-step ideal sub-Riemannian structure with no measure contraction properties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0q5e1rhk5v6w6",
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}