Limiting Carleman weights and conformally transversally anisotropic manifolds
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- by Pablo Angulo, Daniel Faraco, Luis Guijarro and Mikko Salo PDF
- Trans. Amer. Math. Soc. 373 (2020), 5171-5197 Request permission
Abstract:
We analyze the structure of the set of limiting Carleman weights in all conformally flat manifolds, $3$-manifolds, and $4$-manifolds. In particular we give a new proof of the classification of Euclidean limiting Carleman weights, and show that there are only three basic such weights up to the action of the conformal group. In dimension three we show that if the manifold is not conformally flat, there could be one or two limiting Carleman weights. We also characterize the metrics that have more than one limiting Carleman weight. In dimension four we obtain a complete spectrum of examples according to the structure of the Weyl tensor. In particular, we construct unimodular Lie groups whose Weyl or Cotton-York tensors have the symmetries of conformally transversally anisotropic manifolds, but which do not admit limiting Carleman weights.References
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Additional Information
- Pablo Angulo
- Affiliation: Department of Mathematics, ETS de Ingenieros Navales, Universidad Politécnica de Madrid, Spain
- MR Author ID: 919147
- Email: pablo.angulo@upm.es
- Daniel Faraco
- Affiliation: Department of Mathematics, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain; and ICMAT CSIC-UAM-UCM-UC3M, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
- MR Author ID: 704525
- Email: daniel.faraco@uam.es
- Luis Guijarro
- Affiliation: Department of Mathematics, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain; and ICMAT CSIC-UAM-UCM-UC3M, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
- MR Author ID: 363262
- Email: luis.guijarro@uam.es
- Mikko Salo
- Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35, 40014 University of Jyvaskyla, Finland
- MR Author ID: 749335
- Email: mikko.j.salo@jyu.fi
- Received by editor(s): November 28, 2018
- Received by editor(s) in revised form: December 6, 2019
- Published electronically: April 29, 2020
- Additional Notes: The first author was supported by research grant MTM2017-85934-C3-3-P from the Ministerio de Ciencia e Innovación (MCINN) and ERC 301179. The second and third authors were supported by research grants MTM2014-57769-1-P, MTM2014- 57769-3-P, MTM2017-85934-C3-2-P from MCINN, by ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554 (MINECO), by ERC 301179 and by ERC 34728.
The fourth author was supported by the Academy of Finland (grants 284715 and 309963) and by ERC under Horizon 2020 (ERC CoG 770924). - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5171-5197
- MSC (2010): Primary 35R30, 53A30; Secondary 58J32
- DOI: https://doi.org/10.1090/tran/8072
- MathSciNet review: 4127874