{
  "abstract": "The present paper studies integral geometry problems on three-dimensional Riemannian balls, where integration is performed over minimal surfaces or, more generally, $Φ$-surfaces defined by an elliptic curvature functional $Φ$. The space of all $Φ$-surfaces spanned by round circles on the boundary is a three-dimensional manifold, which we call the space of circles. We show that, when the metric is $Φ$-simple - a notion which extends to this setting the notion of simple metrics in the geodesic case -, the Gauss lifts of the $Φ$-surfaces define a foliation of the unit tangent bundle that should be viewed as a two-dimensional analogue of the standard geodesic foliation. This is achieved by solving a foliated Plateau problem on the ball. We then analyze the associated surface Radon transform corresponding to integration along the surfaces and show that it has a finite-dimensional kernel; we also prove that it is injective for an open and dense set of metrics. In the special case of a foliation by minimal surfaces, we apply these results to solve the following boundary area rigidity problem: does the collection of areas of the minimal surfaces determine the metric up to isometry?",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29470",
  "author_total": 4,
  "authors": [
    {
      "name": "Sébastien Alvarez"
    },
    {
      "name": "Thibault Lefeuvre"
    },
    {
      "name": "Ben Lowe"
    },
    {
      "name": "Graham A. Smith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08gc9snn032td.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29470",
  "fulltext_url": null,
  "id": "08gc9snn032td",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29470",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214281867",
  "pdf_url": "https://arxiv.org/pdf/2609.29470",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08gc9snn032td.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29470",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29470"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three",
  "title_text": "Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/08gc9snn032td",
  "venue": "arXiv",
  "versions": []
}