Volume 15, issue 2 (2011)

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On Gromov–Hausdorff stability in a boundary rigidity problem

Sergei Ivanov

Geometry & Topology 15 (2011) 677–697
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.

Keywords
boundary distance rigidity, Gromov–Hausdorff topology
Primary: 53C23