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Marked boundary rigidity for surfaces



We show that, on an oriented compact surface, two sufficiently $C^{2}$ -close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.



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Marked boundary rigidity for surfaces



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