{
  "abstract": "Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is $1$-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition $\\operatorname{CD}(ρ,\\infty)$. A contracting transport map as in these conjectures implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. We construct counterexamples to these conjectured spectral comparisons on spheres in dimensions $d\\geq4$ and on smooth complete weighted manifolds diffeomorphic to $\\mathbb{R}^d$ satisfying the $\\operatorname{CD}(1,\\infty)$ condition in dimensions $d\\geq4$, thereby obtaining obstructions to contracting transport maps. The spherical counterexamples can be chosen arbitrarily close to the unit round metric in $C^\\infty$, while retaining $\\operatorname{Ric_g}\\geq(d-1)g$. The weighted counterexamples can be chosen with non-negative sectional curvature $\\operatorname{Sec}_g\\geq0$ when $d\\geq4$. For $d\\geq5$, we also construct counterexamples satisfying $\\operatorname{Ric}_g\\geq0$ and $\\nabla_g^2V\\geq g$, where $μ=Z^{-1}e^{-V}\\operatorname{dvol}_g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.24705",
  "author_total": 1,
  "authors": [
    {
      "name": "Shrey Aryan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/304vey1c4e4yd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.24705",
  "fulltext_url": null,
  "id": "304vey1c4e4yd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.24705",
  "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.PR"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7162468794",
  "pdf_url": "https://arxiv.org/pdf/2605.24705",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/304vey1c4e4yd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2605.24705",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.24705"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Spectral Obstructions to Contracting Transport Maps on Curved Spaces",
  "title_text": "Spectral Obstructions to Contracting Transport Maps on Curved Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/304vey1c4e4yd",
  "venue": "arXiv",
  "versions": []
}