Boundary rigidity and stability for generic simple metrics

Abstract

We study the boundary rigidity problem for compact Riemannian manifolds with boundary ( M , g ) (M,g) : is the Riemannian metric g g uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρ g ( x , y ) \rho _g(x,y) known for all boundary points x x and y 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 G \mathcal {G} of simple Riemannian metrics such that for any g 0 ∈ G g_0\in \mathcal {G} , any two Riemannian metrics in some neighborhood of g 0 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 G \mathcal {G} .