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Recovering a Riemannian metric from area data

Banff International Research Station for Mathematical Innovation and Discovery

We address a geometric inverse problem: Consider a simply connected Riemannian 3-manifold $(M,g)$ with boundary. Assume that given any closed loop \gamma on the boundary, one knows the area of the area-minimizer bounded by \gamma. Can one reconstruct the metric g from this information We answer this in the affirmative in a very broad open class of manifolds. We will briefly discuss the relation of this problem with the question of reconstructing a metric from lengths of geodesics, and also with the Calderon problem of reconstructing a metric from the Dirichlet-to-Neumann operator for the corresponding Laplace-Beltrami operator. We also raise the analogous question for asymptotically hyperbolic manifolds, and the significance of their question in physics. Joint with T Balehowsky and A Nachman.

Mathematics; Global analysis, analysis on manifolds; Relativity and gravitational theory; Global analysis

video/mp4

Author affiliation: University of Toronto

2019-03-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69047

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/