Boundary rigidity, volume minimality, and minimal surfaces in $L_\infty$: a survey

A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric).
To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can “tap” at some points of the surface of the body and “listen when the sound gets to other points”. The question is if this information is enough to determine what is inside.
This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a “travel time” for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc.
In a joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first class of boundary rigid metrics of non-constant curvature beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one.
The approach is grew up from another long-term project of studying surface area functionals in normed spaces, which we have been working on it for more than ten years. There are a number of related issues regarding area-minimizing surfaces in Riemannian manifold. The talk gives a non-technical survey of ideas involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words, we will not get into any technical details of the proofs).