## Geometry & Topology

### On Gromov–Hausdorff stability in a boundary rigidity problem

Sergei Ivanov

#### 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 $C1$–close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance function of $M$ is $C0$–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.

#### Article information

Source
Geom. Topol., Volume 15, Number 2 (2011), 677-697.

Dates
Received: 27 July 2010
Revised: 24 January 2011
Accepted: 22 February 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732301

Digital Object Identifier
doi:10.2140/gt.2011.15.677

Mathematical Reviews number (MathSciNet)
MR2800362

Zentralblatt MATH identifier
1219.53043

#### Citation

Ivanov, Sergei. On Gromov–Hausdorff stability in a boundary rigidity problem. Geom. Topol. 15 (2011), no. 2, 677--697. doi:10.2140/gt.2011.15.677. https://projecteuclid.org/euclid.gt/1513732301

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