Obstructions to the existence of limiting Carleman weights
  We give a necessary condition for a Riemannian manifold to admit limiting Carleman weights in terms of the Weyl tensor (in dimensions 4 and higher) and the Cotton-York tensor in dimension 3. As an application we provide explicit examples of manifolds without limiting Carleman weights and show that the set of such metrics on a given manifold contains an open and dense set. 
Pablo Angulo-Ardoy, Daniel Faraco, Luis Guijarro and Alberto Ruiz
math.AP