Limiting Carleman weights and conformally transversally anisotropic manifolds

Angulo, Pablo; Faraco, Daniel; Guijarro, Luis; Salo, Mikko

doi:10.1090/tran/8072

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.

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