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 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.

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