{
  "abstract": "We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space $\\mathbb{G}^{n+m}$ is $\\mathbb{R}^{n}\\times\\mathbb{R}^{m}$, equipped with Lebesgue measure and generated by $X_i=\\partial_{x_i}$ and $Y_j=|x|\\partial_{y_j}$, for $1\\leq i\\leq n$ and $1\\leq j\\leq m$. We prove that $\\mathbb{G}^{n+m}$ satisfies $\\operatorname{MCP}(K,N)$ if and only if $N\\geq n+4m$ and $K\\leq 0$. We also show that, for $α\\geq1$, the $α$-Grushin plane generated by $X=\\partial_x$ and $Y_α=|x|^α\\partial_y$ satisfies $\\operatorname{MCP}(K,N)$ if and only if $K\\leq0$ and $N\\geq N_α$, where \\[ N_α:= 1+\\max_{L>1} \\frac{(2α+1)L}{(L-1)^{2α+1}+1}. \\] This resolves the conjecture posed in arXiv:2010.16350 and, for integer $α\\geq2$, provides the first examples of real-analytic sub-Riemannian structures with noninteger curvature exponent. Both results recover the known curvature exponent $5$ of the classical Grushin plane, corresponding respectively to $n=m=1$ and $α=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10208",
  "author_total": 2,
  "authors": [
    {
      "name": "Michael Albert"
    },
    {
      "name": "Samuël Borza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/03db1ecwz9bmg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10208",
  "fulltext_url": null,
  "id": "03db1ecwz9bmg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10208",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212185047",
  "pdf_url": "https://arxiv.org/pdf/2609.10208",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/03db1ecwz9bmg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10208",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10208"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The measure contraction property on Grushin spaces",
  "title_text": "The measure contraction property on Grushin spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/03db1ecwz9bmg",
  "venue": "arXiv",
  "versions": []
}