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Boundary rigidity and stability for generic simple metrics

Authors: Plamen Stefanov and Gunther Uhlmann
Journal: J. Amer. Math. Soc. 18 (2005), 975-1003
MSC (2000): Primary 53C24, 53C20; Secondary 53C21, 53C65
Published electronically: July 5, 2005
MathSciNet review: 2163868
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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 $\mathcal{G}$.

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Additional Information

Plamen Stefanov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

Keywords: Boundary rigidity, Riemannian manifold
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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