Let us continue our posts regarding to the Schur lemma, i.e., the following estimate
holds provided and where is the scalar curvature and is the average of .
Recently, Ge and Wang improved the above inequality for the case . They showed that the above estimate remains valid provided the scalar curvature is non-negative.
Today, we talk about a work by Ezequiel R. Barbosa recently published in Proc. Amer. Math. Soc. 2012 [here]. Following is his main result
Theorem. Let be a -dimensinal closed Riemannian manifold. Then
where is the average of the scalar curvature of and is the Yamabe invariant. Moreover, the equality holds if and only if there exists a metric such that is an Einstein manifold.
As can be seen, the only contribution of the above theorem is to assume no conditions on the Ricci tensor or the scalar curvature.
It is worth noticing that if the Yamabe invariant is nonnegative, then
the above inequality is equivalent to
We now choose the metric in such a way that is constant. Then
since is a Yamabe solution. Making use of , we get
which proves the result.
- Almost-Schur lemma by De Lellis and Topping.
- Almost-Schur lemma on 4-dimensional closed Riemannian manifolds by Ge and Wang.