Boundary rigidity and fillingvolume minimality of metrics close to flat ones
主 题: Boundary rigidity and fillingvolume minimality of metrics close to flat ones
报告人: Prof. D. Burago (Pennsyvania State Univ)
时 间: 2007-11-13 下午 2:00 - 3:00
地 点: 理科一号楼 1479
A compact Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined (upto an isometry) by the distances between the boundarypoints. To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants tofind out what is inside a solid body made of different materials(in other words, properties of the medium change from pointto point). The speed of sound depends on the material. One can "tap" at some points of the surface of the bodyand "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 studies a lot, mainly from the PDE viewpoint. We suggest a completely different approachbased on "minimality". As a matter of fact, we embedour manifolds in a certain normed space and show that they happens tobe a minimal surfaces, and then prove certain uniquenessresults for such surfaces. I will discuss the following result: Euclidean regionswith Riemannian metrics sufficiently close to a Euclidean one are minimal fillings and boundary rigid. This is the firstresult in dim>2 showing boundary rigidity of metrics otherthan extremely special ones (products and symmetric spaces). The talk is based on a joint work with S. Ivanov.