Generalized boundary rigidity and minimal surface transform

TL;DR

This paper investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary, extending boundary rigidity results to higher dimensions and establishing stability using the minimal surface transform.

Contribution

It introduces the minimal surface transform as a generalization of the X-ray transform, proving its invertibility and stability under certain conditions, and extends boundary rigidity results to higher dimensions.

Findings

Proved uniqueness of Riemannian metrics from minimal surface areas in higher dimensions.

Established H"older stability for the inverse problem.

Demonstrated invertibility of the minimal surface transform under foliation conditions.

Abstract

We study a generalized boundary rigidity problem, which investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary. We prove that for a conformal perturbation of an analytic metric in dimension $n+1$ ($n \geq 2$), the metric is determined by these volumes under an ampleness condition. Furthermore, we establish H\"older stability for this determination. This result extends earlier works in dimension $2+1$. Instead of relying on reductions to Calder\'on type problems and complex geometrical optics solutions, we study the linearized forward operator that gives rise to the minimal surface transform, a generalization of the X-ray/Radon transform. We demonstrate that this transform fits into the framework of double fibration transforms and satisfies the Bolker condition in the sense of Guillemin. Under certain assumptions, including a foliation condition, we prove invertibility of this transform on an analytic manifold as well as recovery of the analytic wave front set. The methods developed in this paper offer new tools for addressing the generalized boundary rigidity problem and expand the scope of applications of double fibration transforms. We anticipate that these techniques will also be applicable to other geometric inverse problems. Beyond mathematics, our results have implications for the AdS/CFT correspondence in physics.