Limiting Carleman weights and conformally transversally anisotropic manifolds

TL;DR

This paper classifies and analyzes the structure of limiting Carleman weights across various conformally flat and non-flat manifolds, providing new proofs and characterizations in dimensions three and four.

Contribution

It offers a new proof of Euclidean limiting Carleman weights classification and characterizes metrics with multiple weights in 3D and 4D, expanding understanding of their geometric structure.

Findings

Only three basic Euclidean limiting Carleman weights exist up to conformal group actions.

Non-conformally flat 3-manifolds may have one or two limiting Carleman weights.

Constructed examples of Lie groups with specific Weyl tensor symmetries lacking limiting Carleman weights.

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.