Boundary rigidity and volume minimality for metrics close to a flat
one
Dmitri Burago
Pennsylvania State
October 30, 2006
Abstract: 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 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
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.