Marked boundary rigidity for surfaces of Anosov type
Abstract
Let $\Sigma$ be a smooth compact connected oriented surface with boundary. A metric on $\Sigma$ is said to be of Anosov type if it has strictly convex boundary, no conjugate points, and a hyperbolic trapped set. We prove that two metrics of Anosov type with the same marked boundary distance are isometric (via a boundarypreserving isometry isotopic to the identity). As a corollary, we retrieve the boundary distance rigidity result for simple disks of Pestov and Uhlmann [arXiv:math/0305280]. The proof rests on a new transfer principle showing that, in any dimension, the marked length spectrum rigidity conjecture implies the marked boundary distance rigidity conjecture under the existence of a suitable isometric embedding into a closed Anosov manifold. Such an isometric embedding result for open surfaces of Anosov type was proved by the first author with Chen and Gogolev in [arXiv:2009.13665] while the marked length spectrum rigidity for closed Anosov surfaces was established by the second author with Guillarmou and Paternain in [arXiv:2303.12007].
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.06893
 arXiv:
 arXiv:2305.06893
 Bibcode:
 2023arXiv230506893E
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 Mathematics  Dynamical Systems;
 53C24;
 53C22;
 37C27;
 37D40;
 57K20;
 32G20
 EPrint:
 21 pages, 2 figures