Calculus of Variations and Geometric Measure Theory

C. De Lellis - Peter M. Topping

Almost-Schur lemma

created by delellis on 06 May 2011
modified on 27 Jun 2019


Published Paper

Inserted: 6 may 2011
Last Updated: 27 jun 2019

Journal: Calc. Var.
Volume: 49
Pages: 347-354
Year: 2012

ArXiv: 1003.3527 PDF

For the update version and eventual errata see the webpage http:/www.math.uzh.chdelellis


Schur's lemma states that every Einstein manifold of dimension $n\geq 3$ 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.