Almost-Schur lemma

摘要

Schur’s lemma states that every Einstein manifold of dimension n ≥ 3 has constant scalar curvature. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be small rather than identically zero. In particular, we provide an optimal L 2 estimate under suitable assumptions and show that these assumptions cannot be removed.