2023/05/11 by Alena Erchenko, Erchenko, Alena, Thibault Lefeuvre +1
Computer Science · Mathematics · #32G20 #37C27 #37D40 #53C22 #53C24 #57K20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2305.06893
openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Σ be a smooth compact connected oriented surface with boundary. A metric on Σ 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 boundary-preserving 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].