Riemannian Anosov extension and applications

Dong Chen; Alena Erchenko; Andrey Gogolev

doi:10.5802/jep.237

Riemannian Anosov extension and applications
[Extension d’Anosov riemannienne et applications]

Dong Chen ¹ ; Alena Erchenko ² ; Andrey Gogolev ¹

¹ Department of Mathematics, The Ohio State University 100 Math Tower, 231 W 18th Ave, Columbus, OH 43210, USA
² Department of Mathematics, The University of Chicago Eckhart Hall, 5734 S University Ave, Chicago, IL 60637, USA

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 945-987.

Résumé
Abstract

Soit $Σ$ une variété riemannienne avec bord sphérique strictement convexe. Lorsque la métrique n’a pas de points conjugués et que l’ensemble capté est hyperbolique, nous montrons que $Σ$ peut être plongée isométriquement dans une variété riemannienne fermée dont le flot géodésique est Anosov. Nous utilisons ce plongement pour établir un lien direct entre le théorème de Livshits classique pour les flots d’Anosov et le théorème de Livshits pour la transformée en rayons X qui apparaît dans le programme de rigidité des bords. Nous donnons également une application pour la rigidité lenticulaire dans une classe conforme.

Let $Σ$ be a Riemannian manifold with strictly convex spherical boundary. Assuming absence of conjugate points and that the trapped set is hyperbolic, we show that $Σ$ can be isometrically embedded into a closed Riemannian manifold with Anosov geodesic flow. We use this embedding to provide a direct link between the classical Livshits theorem for Anosov flows and the Livshits theorem for the X-ray transform which appears in the boundary rigidity program. Also, we give an application for lens rigidity in a conformal class.

Reçu le : 2021-08-04
Accepté le : 2023-04-25
Publié le : 2023-05-11

DOI : 10.5802/jep.237

Classification : 37D40, 37D20, 53C24, 53C21
Keywords: Anosov flow, geodesic flow, lens rigidity, Livshits theorem, trapped sets
Mot clés : Flot Anosov, flot géodésique, rigidité lenticulaire, théorème de Livshits, ensemble capté

Affiliations des auteurs :

Dong Chen ¹ ; Alena Erchenko ² ; Andrey Gogolev ¹

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Riemannian {Anosov} extension and applications},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {945--987},
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Dong Chen; Alena Erchenko; Andrey Gogolev. Riemannian Anosov extension and applications. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 945-987. doi : 10.5802/jep.237. https://jep.centre-mersenne.org/articles/10.5802/jep.237/

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