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

TL;DR

This work addresses boundary and lens rigidity for 2D Riemannian manifolds with boundary by proving local rigidity near convex boundary, global rigidity for non-trapping convex-boundary settings (affirming Uhlmann's conjecture), and optimal rigidity with trapping up to outermost geodesics. The authors develop a nonlinear, reconstruction-driven framework that recasts lens data as circlefronts and pseudo-circlefronts and studies their evolution via Jacobi fields, transporting these objects through Gaussian coordinates and a newly introduced thermostatic vector field. They derive a coupled system of PDEs and transport equations for the circlefront differences, then obtain energy-type bounds through Hardy-type estimates, a convex foliation zλ, and Volterra-type equations that tie the two-variable metric perturbations to boundary data. The novelty lies in avoiding linearization, directly controlling nonlinear circlefront dynamics, and proving rigidity via an energy method that accommodates trapping, thereby extending Pestov–Uhlmann and X-ray transform results to broader two-dimensional geometries with convex boundaries. The techniques offer a pathway toward stability estimates and practical reconstruction algorithms for the boundary/radius data problem in 2D. The results have significance for travel-time tomography and related geophysical imaging, where boundary measurements determine interior wave speeds encoded by the metric.

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.

Figures (6)

• Figure 1: LEFT IMAGE: A fan of geodesics (in red) emanating from a point $A$ in $M$, in (green) directions $\vec{\nu}(\theta)$ all of which intersect the boundary $\partial M$ at points $p(\theta,A)$. The "fan" of angles $\theta$ has "opening" $\pi$. RIGHT FIGURE: Dual image of the left. The vector field ${(}p(A;\theta), v(A,\theta){)}$ (in green) along a subinterval in $\partial M$. (The vectors in green are $v(A,\theta)$ at $p(\theta; A)$). Here the "fan" $\theta$ has angle $\pi$. The red lines are the geodesics $\gamma_{p(A;\theta), v(A,\theta)}$, which all focus at the point $A$.
• Figure 2: The extended scattering map, on $\widetilde{M}\setminus M$: The black curves are geodesic segments in the "known manifold" $\widetilde{M}$. The red points are the entry points of these geodesic segments into the "unknown" $M$. The dotted lines are the portions of these geodesics in the "unknown" $M$.
• Figure 3: The coordinate system $x,y$: $y$ level sets in light blue and $x$ level sets in purple. The horizontal boundary $\Gamma_h=\{x=0\}$ and the vertical boundaries $\Gamma_v=\{y=\pm \delta\}$ are in green. Note $\Gamma=\Gamma_v\bigcup \Gamma_h\subset \widetilde{M}\setminus M$.
• Figure 4: The (red) vectors $\vec{\xi}(x,y,\theta)$ based at $(x,y)$ for different angles $\theta: 0<\theta_1<\frac{\pi}{2}<\theta_2$. The (purple) lines are the corresponding geodesic segments $\gamma_{\phi(x,y), \xi(x,y,\theta)}(t)$ emanating from $\vec{\xi}(x,y,\theta)$ for negative times. At $t=-T$ the corresponding vectors $\vec{\Psi}^2(x,y,\theta, -T)$ at $t=-T$ are highlighted. The green curve is $\bigcup_{\theta\in [0,\pi]}\Psi^1(x,y,\theta, -T)$. Note that $\bigcup_{\theta\in [0,\pi]}\Psi^1(x,y,\theta, -T)\subset \widetilde{M}\setminus M$.
• Figure 5: The condition (A1): The values $\vec{\Psi}(x,y,\theta, -T)$ are required to agree with the scattering relation $S^{-T}_{\widetilde{M},M}(\vec{\Psi}_0(x,y,\theta))$, to ensure that $\vec{\Psi}(x,y,\theta, 0)$ agrees with $\vec{\Phi}(x,y,\theta)$. This imposition thus "sees" the lens data, since $S^{-T}_{\widetilde{M},M}$ is built out of that lens data, via the map defined by the dotted lines: The black curves are geodesic segments in the "known manifold" $\widetilde{M}$. The red points are the entry points of these geodesic segments into the "unknown" $M$. The dotted lines are the portions of these geodesics in the "unknown" $M$.

Theorems & Definitions (112)

• Theorem 1
• Remark 1.1
• Theorem 2
• Theorem 3
• Definition 2.1.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.2.1
• Remark 2.4