Generalized boundary rigidity and minimal surface transform

TL;DR

This work addresses generalized boundary rigidity by asking whether the areas of embedded minimal surfaces determine a Riemannian metric with boundary under perturbations. The authors linearize the area measurements to obtain the minimal surface transform, a Radon-type operator that fits into the double fibration framework and satisfies the Bolker condition, enabling microlocal inversion. They prove invertibility and Hölder stability under ampleness or foliation conditions on an analytic background metric, constructing finite-dimensional subfamilies of minimal surfaces and Dirichlet data to achieve global ellipticity and stable recovery of the conformal factor. The results extend prior 2+1 dimensional theory to higher dimensions, provide new tools for double fibration transforms, and have potential implications for bulk reconstruction in the AdS/CFT correspondence, offering robust stability where previous Calderón-type approaches faced limitations.

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 Hölder stability for this determination. This result extends earlier works in dimension $2+1$. Instead of relying on reductions to Calderón 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.

Theorems & Definitions (81)

• Definition 1.1
• Theorem 1.2: Informal statement; see Theorem \ref{['thm:stability']} for precise statement
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Definition 2.1: Admissible minimal surface
• Theorem 2.2
• Proposition 2.3
• Lemma 2.4
• proof