Title: Boundary rigidity of Riemannian manifolds
Speaker: Professor Plamen Stefanov
Speaker Info: Purdue
Brief Description:
Special Note:
Abstract:
This talk is based on a series of joint works with Gunther Uhlmann. Let $(M,g)$ be a compact Riemannian manifold with boundary. Assume that for each pair of boundary points (x,y), we know the distance function $d(x,y)$. The manifold $M$ is called boundary rigid, if it is determined uniquely by $d(x,y)$ (known on the boundary). At present, boundary rigidity is known only for some classes of manifolds.Date: Thursday, February 5, 2004We study this problem for simple manifolds, i.e., strictly convex ones with no caustics inside. The associated linearized problem is recovery of the so-called solenoidal part of a tensor from its X-ray transform $If$ along geodesics. We show that $N=I^*I$ is a pseudodifferential operator, analyze its principal symbol and invert it microlocally. We prove an a priori stability estimates for $N$. Using analytic pseudodifferential calculus, we show that for analytic metrics, the linearized problem is invertible. For the non-linear one, we prove generic local uniqueness and stability.