Almost-Schur lemma

Abstract

Schur's lemma states that every Einstein manifold of dimension $n\ge 3$ has constant scalar curvature. Here $(M,g)$ is defined to be Einstein if its traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be \emph{small} rather than identically zero.

Remarks and references added. To appear in Calc. Var. See also http://www.math.uzh.ch/delellis