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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, notably those that admit sweep-outs by minimal surfaces from all directions. 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. Connections with this question in the AdS-CFT correspondence will also be made. Joint with T. Balehowsky and A. Nachman.

Mathematics; Global analysis, analysis on manifolds; Dynamical systems and ergodic theory

video/mp4

Author affiliation: University of Toronto

2020-09-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76012

Unreviewed

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