On the set of metrics without local limiting Carleman weights

TL;DR

This paper proves that the set of Riemannian metrics lacking local limiting Carleman weights at any point is both open and dense, extending previous results on the genericity of such metrics using conformal invariants.

Contribution

It establishes that the set of metrics without local limiting Carleman weights at any point is open and dense, advancing understanding of metric properties related to Carleman weights.

Findings

The set of metrics without local limiting Carleman weights is open.

The set of metrics without local limiting Carleman weights is dense.

This set includes metrics with no local conformal diffeomorphisms between open subsets.

Abstract

In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This paper is a continuation of arXiv:1411.4887 [math.AP] in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights \emph{at any point} is open and dense.