Marked boundary rigidity for surfaces

TL;DR

This paper proves that on certain compact surfaces, if two Riemannian metrics have the same marked boundary distance and satisfy specific geometric conditions, then they are isometric via a boundary-fixing diffeomorphism, extending previous results.

Contribution

It extends marked boundary rigidity results to a broader class of surfaces, including those with hyperbolic trapped sets and negative curvature, under less restrictive conditions.

Findings

Metrics with same marked boundary distance are isometric under given conditions.

Extension of Croke and Otal's result to negatively curved surfaces.

Validation of rigidity for metrics close in the $C^2$ topology.

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 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 same marked boundary distance, extending a result of Croke and Otal.