Forum of Mathematics, Sigma (Jan 2017)

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF LIMITING CARLEMAN WEIGHTS

  • PABLO ANGULO-ARDOY,
  • DANIEL FARACO,
  • LUIS GUIJARRO

DOI
https://doi.org/10.1017/fms.2017.4
Journal volume & issue
Vol. 5

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.

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