On Gromov–Hausdorff stability in a boundary rigidity problem

Ivanov, Sergei

doi:10.2140/gt.2011.15.677

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On Gromov–Hausdorff stability in a boundary rigidity problem

Sergei Ivanov

Geometry & Topology 15 (2011) 677–697

DOI: 10.2140/gt.2011.15.677

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

Mathematical Subject Classification 2000

Primary: 53C23

References

Publication

Received: 27 July 2010

Revised: 24 January 2011

Accepted: 22 February 2011

Published: 10 May 2011

Proposed: Dmitri Burago

Seconded: Leonid Polterovich, Jean-Pierre Otal

Authors

Sergei Ivanov
St.Petersburg Department Steklov Mathematical Institute RAS 27, Fontanka 191023 St Petersburg Russia
http://eimi.imi.ras.ru/eng/perso/svivanov.php

Volume 15, issue 2 (2011)