Obstructions to the existence of limiting Carleman weights

TL;DR

This paper establishes necessary conditions involving the Weyl and Cotton-York tensors for the existence of limiting Carleman weights on Riemannian manifolds, providing explicit examples and showing such metrics are common.

Contribution

It introduces new tensor-based criteria for limiting Carleman weights and demonstrates their non-existence on a generic set of manifolds.

Findings

Manifolds without limiting Carleman weights are explicitly constructed.

The set of metrics lacking limiting Carleman weights is open and dense.

Necessary conditions involve the Weyl tensor in dimensions 4 and higher, and the Cotton-York tensor in dimension 3.

Abstract

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.