A compact Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined (up to an isometry) by the distances between the boundary points.
To visualize this, imagine wanting to find out what the Earth is made of, or, more generally, 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 whether this information is enough to determine what is inside.
This problem has been studied a lot, mainly from PDE viewpoint. We suggest an alternative approach based on “minimality”. A manifold is said to be a minimal filling if it has the least volume among all compact (Riemannian) manifolds with the same boundary and the same or greater boundary distances.
I will discuss the following result: Euclidean regions with Riemannian metrics sufficiently close to a Euclidean one are minimal fillings and boundary rigid. This is the first result proving that in dim>2 metrics other than extremely special ones (of constant curvature) are rigid. The talk is based on a joint work with S. Ivanov