Boundary
rigidity of metrics close to a flat one
Professor Dmitri Burago
Department of Mathematics
The
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 studies 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 in dim>2 showing boundary rigidity
of metrics other than extremely special ones (products and symmetric
spaces). The talk is based on a joint work with S. Ivanov
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Room 517, Meng Wah Complex, HKU |
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All are welcome |
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