Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three

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?