Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

TL;DR

This paper identifies a flaw in a previous estimate for skew-symmetric 2-tensors, providing a counterexample that challenges its use in proving classification results like the Black Hole Uniqueness Theorem.

Contribution

It presents a counterexample to a previously claimed divergence lower bound for covariant derivatives of skew-symmetric 2-tensors, questioning prior applications in geometric analysis.

Findings

Counterexample invalidates the previous estimate.

Challenges the proof of the Black Hole Uniqueness Theorem.

Questions the validity of certain classification results in geometric analysis.

Abstract

In arXiv:2103.15482 an estimate for suitable skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature and in connection with the Besse Conjecture. In the present note we point out an issue in the argument proposed in arXiv:2103.15482 and we provide a counterexample to the estimate.