Let us continue our previous post regarding to the Schur lemma, i.e., the following estimate

holds provided and where is the scalar curvature and is the average of .

It was proved by De Lellis and Topping that the condition cannot be relaxed. Also, the condition plays an important role in their argument.

Very recently, in their paper, Ge and Wang proved the following

**Theorem**. If and if is a closed Riemannian manifold with nonnegative scalar curvature, then

olds. Moreover, equality holds if and only if is an Einstein manifold.

Also, if we denote by the -scalar curvature of metric , they found that the above inequality is equivalent to the following

As such, instead of proving the former inequality, they aimed to prove the latter one. In order to mention their proof, let us recall the definition of the -scalar curvature, which was first introduced by Viaclovsky in his PhD thesis and has been intensively studied by many mathematicians.

Let

be the Schouten tensor of . For an integer with let be the -th elementary symmetric function in . The -scalar curvature is

where is the set of eigenvalues of the matrix . In particular, and .

Clearly,

and

In order to prove the inequality, they proved the following

**Lemma**. For any and any closed Riemannian manifold , thereexists a conformal metric satisfying

where the term on the right hand side is usually called the Yamabe invariant, i.e.,

The proof of the lemma depends on the following: for any symmetric matrix , there holds

The metric is chosen in such a way that the scalar curvature of is constant. This is the so-called Yamabe problem.

In order to prove the main result, they observed that in the case of dimension , it is well known that is constant in any given conformal class. Therefore,

Recently, Ye and Wang also extended the above result to the case . We shall discuss this result later.

See also:

- Almost-Schur lemma by De Lellis and Topping.

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