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SUFFICIENT CONDITIONS FOR THE EXISTENCE OF LIMITING CARLEMAN WEIGHTS

Part of: Classical differential geometry Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  02 March 2017

PABLO ANGULO-ARDOY ,
DANIEL FARACO  and
LUIS GUIJARRO
PABLO ANGULO-ARDOY
Affiliation:
E.T.S de Ingenieros Navales, Universidad Politécnica de Madrid, Madrid, Spain; pablo.angulo@uam.es
DANIEL FARACO
Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, Madrid, Spain ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain; daniel.faraco@uam.es, luis.guijarro@uam.es
LUIS GUIJARRO
Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, Madrid, Spain ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain; daniel.faraco@uam.es, luis.guijarro@uam.es

Abstract

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In Angulo-Ardoy et al. [Anal. PDE, 9(3) (2016), 575–596], we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based on the Cotton–York tensor in dimension 3 and the Weyl tensor in dimension 4. In this paper, we find further necessary conditions for the existence of local LCWs that are often sufficient. For a manifold of dimension 3 or 4, we classify the possible Cotton–York, or Weyl tensors, and provide a mechanism to find out whether the manifold admits local LCW for each type of tensor. In particular, we show that a product of two surfaces admits an LCW if and only if at least one of the two surfaces is of revolution. This provides an example of a manifold satisfying the eigenflag condition of Angulo-Ardoy et al. [Anal. PDE, 9(3) (2016), 575–596] but not admitting LCW.

MSC classification

Primary: 35R30: Inverse problems
Secondary: 53A30: Conformal differential geometry
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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