Boundary rigidity and stability for generic simple metrics

Stefanov, Plamen; Uhlmann, Gunther

doi:10.1090/S0894-0347-05-00494-7

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 $\mathcal {G}$.

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