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Descripción
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| Two variants of the Calderón Inverse Problems in the presence of an unisotropic ambient metric can be solved if that metric admits one or more Limiting Carleman Weights (LCWs), as shown by Dos Santos Ferreira-Kenig-Salo-Uhlmann in 2006. In that paper, they also showed that the existence of LCW for a given metric is roughly equivalent to the metric being Conformal Transversally Anisotropic: the metric is a conformal multiple of a product metric M = RxN. We find necessary conditions for a metric to be CTA in terms of the classical conformally invariant tensors: the Cotton and the Weyl tensors. We find a new proof that the set of metrics that do not admit any local LCW is open and dense. Our necessary conditions also provide some candidate directions for the "R factor" in the splitting RxN, which we can exploit in order to decide whether a given manifold is CTA. We derive from this idea a procedure that can often find the conformal product structure in a given CTA metric. We study products of surfaces and Lie groups with invariant metrics, which provide examples of manifolds whose Weyl or Cotton tensors satisfy the necessary conditions, but may not be CTA. We study manifolds that admit two or more orthogonal LCWs. Finally, we give a new proof of the classification of LCWs in R^n, and study the action of the group of conformal transformations on the set of LCWs of R^n. | |
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Internacional
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Si |
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Nombre congreso
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Sinergy of Inverse Problems at Bilbao |
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Tipo de participación
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960 |
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Lugar del congreso
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BCAM |
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Revisores
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Si |
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ISBN o ISSN
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0000-0000 |
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DOI
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Fecha inicio congreso
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26/06/2019 |
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Fecha fin congreso
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26/06/2019 |
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Desde la página
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1 |
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Hasta la página
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33 |
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Título de las actas
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http://www.bcamath.org/en/courses/2019-06-26-bcam-course/general |