Sufficient conditions for the existence of limiting Carleman weights

In https://arxiv.org/abs/1411.4887, we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based upon 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 a 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 \cite{AFGR} but not admitting $LCW$.