Stability of boundary distance representation and reconstruction of Riemannian manifolds (Q995443)

Embedding an abstract Riemannian manifold into a function space is an important tool to study the Riemannian manifold, for example, convergence and metric properties. For a manifold \(M\) with boundary, a natural method is to embed the manifold into the space \(C(\partial M)\) of continuous functions on the boundary \(\partial M\) via the distance function to the interior point \(r_x\), \(x\in M\): for every \(z\in \partial M\), \(r_x(z)=d(x,z)\). Such embedding occurs naturally in the heat conduction and other contexts. An important problem is to recover a manifold from such a representation. This is closely related to the stability of the embedding. In practice, the data from measurement are often approximate, such a stability property is crucial. The purpose of this paper is to show that under some geometric bound on the manifold and the boundary, the embedding is stable, and furthermore, given a finite discrete set of approximate boundary distance functions, the paper constructs a finite metric space that approximates the Riemannian manifold \(M\) in the sense of Gromov-Hausdorff convergence.