Boundary rigidity and stability for generic simple metrics

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 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 $\mathcal{G}$.