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

Abstract

In [3] 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 [6] and in connection with the Besse Conjecture [8]. In the present note we point out an issue in the argument proposed in [3] and we provide a counterexample to the estimate.