## Boundary rigidity and stability for generic simple metrics

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- by Plamen Stefanov and Gunther Uhlmann
- J. Amer. Math. Soc.
**18**(2005), 975-1003 - DOI: https://doi.org/10.1090/S0894-0347-05-00494-7
- Published electronically: July 5, 2005
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## 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 $\mathcal {G}$.## References

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## Bibliographic 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