{
  "abstract": "We study the stability of optimal transport for the squared Carnot--Carathéodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic $L^2$ stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent $0<α<1/2$ on two-generating structures, and the exponent $1/2$ in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to $W_1$ cannot exceed $1/4$ and $1/2$, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order $t$ in $W_1$ produces map variation of order $t^{1/s}$ in an equiregular region of step $s$. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09688",
  "author_total": 2,
  "authors": [
    {
      "name": "Bang-Xian Han"
    },
    {
      "name": "Zhuo-Nan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fbrc0zmvmxga.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09688",
  "fulltext_url": null,
  "id": "9fbrc0zmvmxga",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09688",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7220918656",
  "pdf_url": "https://arxiv.org/pdf/2610.09688",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fbrc0zmvmxga.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09688",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09688"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Stability of optimal transport on sub-Riemannian manifolds",
  "title_text": "Stability of optimal transport on sub-Riemannian manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/9fbrc0zmvmxga",
  "venue": "arXiv",
  "versions": []
}