On Gromov-Hausdorff stability in a boundary rigidity problem

TL;DR

This paper demonstrates that a compact Riemannian manifold with boundary is close to a convex Euclidean region in the Gromov-Hausdorff sense if their boundary distance functions are sufficiently close, under various regularity and volume conditions.

Contribution

It extends boundary rigidity results by establishing Gromov-Hausdorff stability under weaker boundary distance function closeness and volume constraints.

Findings

Manifold is Gromov-Hausdorff close to convex Euclidean region when boundary distance functions are $C^1$-close.

Stability result holds under $C^0$-closeness of boundary distance functions with volume and ball volume lower bounds.

Provides quantitative estimates linking boundary data closeness to geometric proximity.

Abstract

Let $M$ be a compact Riemannian manifold with boundary. We show that $M$ is Gromov-Hausdorff close to a convex Euclidean region $D$ of the same dimension if the boundary distance function of $M$ is $C^1$-close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance function of $M$ is $C^0$-close to that of $D$, the volumes of $M$ and $D$ are almost equal, and volumes of metric balls in $M$ have a certain lower bound in terms of radius.