Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three
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?
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BibTeX
@misc{alvarez2026foliated,
title = {{Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three}},
author = {Sébastien Alvarez and Thibault Lefeuvre and Ben Lowe and Graham A. Smith},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29470},
eprint = {2609.29470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29470},
}
RIS
TY - UNPB
TI - Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three
AU - Sébastien Alvarez
AU - Thibault Lefeuvre
AU - Ben Lowe
AU - Graham A. Smith
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29470
UR - https://arxiv.org/abs/2609.29470
AB - 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?
ER -