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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Boundary rigidity and stability for generic simple metrics
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by Plamen Stefanov and Gunther Uhlmann PDF
J. Amer. Math. Soc. 18 (2005), 975-1003 Request permission

Abstract:

We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $\rho _g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper local and global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set $\mathcal {G}$ of simple Riemannian metrics such that for any $g_0\in \mathcal {G}$, any two Riemannian metrics in some neighborhood of $g_0$ having the same distance function, must be isometric. Similarly, there is a generic set of pairs of simple metrics with the same property. We also prove Hölder type stability estimates for this problem for metrics which are close to a given one in $\mathcal {G}$.
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Additional Information
  • Plamen Stefanov
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • MR Author ID: 166695
  • Email: stefanov@math.purdue.edu
  • Gunther Uhlmann
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • MR Author ID: 175790
  • Email: gunther@math.washington.edu
  • Received by editor(s): January 20, 2005
  • Published electronically: July 5, 2005
  • Additional Notes: The first author was partly supported by NSF Grant DMS-0400869
    The second author was partly supported by NSF and a John Simon Guggenheim fellowship
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 18 (2005), 975-1003
  • MSC (2000): Primary 53C24, 53C20; Secondary 53C21, 53C65
  • DOI: https://doi.org/10.1090/S0894-0347-05-00494-7
  • MathSciNet review: 2163868