Boundary rigidity and stability for generic simple metrics

Details Boundary rigidity and stability for generic simple metrics Boundary rigidity and stability for generic simple metrics

2004-08-05

arxiv.org/abs/math/0408075v1

Mathematics

Differential Geometry

Scientific paper

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 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 and an open dense set $\mathcal{U}\subset \mathcal{G}\times\mathcal{G}$, such that any two Riemannian metrics in $\mathcal{U}$ having the same distance function, must be isometric. We also prove H\"older type stability estimates for this problem for metrics which are close to a given one in $\mathcal{G}$.

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