Lens rigidity in 2D: The reconstruction of a Riemann surface from its geodesic lengths

TL;DR

This paper proves new local and global boundary and lens rigidity results for 2D Riemannian manifolds, including cases with trapping, extending classical work and confirming a conjecture of Uhlmann.

Contribution

It establishes the first comprehensive nonlinear rigidity results for 2D Riemannian manifolds with convex boundary, including trapping scenarios, using novel energy estimates.

Findings

Proves local boundary rigidity near convex boundary portions.

Establishes global rigidity for convex boundary without trapping.

Shows optimal reconstruction even with trapping, confirming Uhlmann's conjecture.

Abstract

We address the question of whether a Riemannian manifold-with-boundary (M,g) in dimension two is uniquely determined from knowledge of the distances between points on its boundary. An affirmative answer is called boundary rigidity for (M,g); it is closely related to lens rigidity. The latter question originates in the problem of reconstructing the speed of sound in an unknown medium from measurements of the travel time of sound waves that are sent in and ultimately return to the boundary. We prove essentially optimal results on these rigidity questions: Our first result answers proves rigidity locally, near a convex portion of the boundary. Our second result proves rigidity globally, for manifolds with convex boundary, in the absence of trapping (closed geodesics), thus confirming a conjecture of Uhlmann. Our final result proves the optimal reconstruction for convex boundaries even in the presence of trapping, showing rigidity up to outermost trapped geodesics. Our results thus extend the classical work of Pestov and Uhlmann on rigidity of simple 2-manifolds, as well as the many prior results on injectivity of the X-ray transform, which address linearized versions of the rigidity problem. \par Our method is to treat the (non-linear) rigidity problem directly, where we simultaneously re-cast the lens data as generalized Riemannian circles, and obtain rigidity for these ``pseudo-circles'', by studying a system of equations that we show these objects must satisfy. The rigidity we obtain ultimately is proven via {novel} estimates that are reminiscent of energy-type estimates for hyperbolic equations.