DIFFERENTIAL GEOMETRY/PDE SEMINAR

Wednesday, May 28, 2003 Padelford C-36 3:45 pm

Microlocal approach to boundary rigidity of Riemannian manifolds

Plamen Stefanov (Purdue University) )

In this talk I will present a recent result by me and Gunther Uhlmann devoted to the boundary rigidity problem for domains in R^{n}: is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function d_{g}(x, y) known for all boundary points x and y? It was conjectured by Michel that this was true for simple metrics. We study the linearized problem first which consists of determining a symmetric 2tensor, up to a potential term, from its geodesic X-ray integral transform I_{g}. We prove that the normal operator N_{g} = I_{g}^{∗}I_{g} is a pseudodifferential operator provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove hypoelliptic type of stability estimate related to the linear problem. Next we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the non-linear boundary rigidity problem near that g.

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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