From Asymptotic Geometry of Tori to Boundary Rigidity: A Survey

Dmitri Burago
Penn State

Imagine that one wants to find out what the Earth is made of. The speed of sound is different in different minerals. One can "tap" at some points on the surface of the Earth and "listen when the sound gets to other points". The question is whether this information is enough to determine what is inside.

More formally, we have a region and a (Riemannian) metric inside it. We can measure distances between boundary points: they form the boundary distance function. One says that it is rigid if it uniquely determines the metric inside the region.

In a joint work with S. Ivanov we prove that if a metric has small second derivatives then it is boundary rigid. This is the first result proving boundary rigidity in Dim2 for metrics other than extremely special ones (of constant curvature). The methods grew from our earlier work on asymptotic geometry of (covers of) tori and study of minimal surfaces in normed spaces. This talk is a survey of this work.