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Marked boundary rigidity for surfaces

Published online by Cambridge University Press:  08 November 2016

COLIN GUILLARMOU
Affiliation:
DMA, UMR 8553 CNRS, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France email cguillar@dma.ens.fr
MARCO MAZZUCCHELLI
Affiliation:
UMPA, UMR 5669 CNRS, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France email marco.mazzucchelli@ens-lyon.fr

Abstract

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.

Type
Original Article
Copyright
© Cambridge University Press, 2016 

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