A note on the almost-Schur lemma on smooth metric measure spaces

In this paper, we prove almost-Schur inequalities on closed smooth metric measure spaces, which implies the results of Cheng and De Lellis–Topping whenever the weighted function f is constant.


Introduction
In 2012, De Lellis and Topping [11] proved an almost-Schur lemma; that is, if a closed Riemannian manifold has nonnegative Ricci curvature, an almost-Schur inequality involves scalar curvature and Ricci curvature: (1.1) In particular, the equality holds if and only if this manifold is Einstein and has constant scalar curvature. In [9], Ge and Wang proved the almost-Schur lemma under the condition of nonnegative scalar curvature in a four-dimensional Riemannian manifold.
In this paper, we study De Lellis-Topping type inequality on a smooth metric measure space. First, we recall some definitions of smooth metric measure space.
For an n-dimensional closed Riemannian manifold (M n , g) and a smooth function f on M, a triple (M n , g, dv f ) is a smooth metric measure space with a weighted volume identity dv f = e -f (x) dv, where dv is the volume element of M with respect to the metric g. Let (∇f ⊗ ∇u) ij = 1 2 (f ,i u ,j + f ,j u ,i ), and let Hess be the Hessian under the metric g. We define the weighted Laplacian by the trace of (Hess f u) ij ≡ (Hess u) ij -(∇f ⊗ ∇u) ij ; that is,

Results and discussion
First, we show the work by Wu [18], which is a type of inequality for an almost-Schur lemma on smooth metric measure spaces. Let Thus, he generalized De Lellis and Topping's result. From Wu's work, we want to improve the inequality that is an expansion of the almost-Schur inequality (1.1) for more general Ricci curvature conditions.
In this paper, for convenience, unless otherwise specified, we provide some notation as follows: Now we state our results.
and λ 1 is the first positive eigenvalue of the weighted Laplacian f . Moreover, the equality holds if and only if M is Einstein and has constant scalar curvature with respect to the metric g.
and λ 1 is the first positive eigenvalue of the weighted Laplacian f .
Remark 2.1 Inequality (2.1) in Theorem 2.1 is sharp in the sense of two aspects. One is that the constant is equal to the square root of the constant in inequality (1.2), then this inequality implies inequality (1.2) whenever f tends to a constant. The other is that if the equality of (2.1) holds, then M is Einstein and has constant scalar curvature with respect to the metric g.
Remark 2.2 In Theorem 2.3, inequality (2.3) is almost the same as inequality (1.2). If the equality of (2.3) holds, "M is trivial Einstein and has constant scalar curvature" remains an open problem. We also note that, due to the work of Cheng [6], we have a partial result about this topic (see Sect. 2.2).

Proofs of Theorems and 2.2
First, it is easy to verify that in Theorems 2.1, 2. Proof of Theorem 2.1 Assume that R is the nontrivial scalar curvature on M with respect to metric g, and R f = R + f . According to the Sobolev embedding theorem and calculus variation, there exists a nontrivial solution u : M → R of the equation We also note that the second Bianchi identity div Ric = 1 2 ∇R implies That is, where whenever Ric f ≥ ( f -(n -1)K)g. Since the first positive eigenvalue λ 1 (see [1,8,12]) of the weighted Laplacian on M is characterized by for which it gives the inequalities Now, by (2.10), we may rewrite (2.6) as which implies the De Lellis-Topping type inequality Ric f -Hess f L 2 + f L 2 . (2.11) If the equality of (2.11) holds, we have the following properties: (i) Ric f (∇u, ·) = ( f -(n -1)K)g(∇u, ·); (ii) μ 1 (Ric f -Hess f ) = Hess f u -f u n g, where μ 1 is a non-zero constant; (iii) R f -R f = -λ 1 u = μ 2 f , where μ 2 is a non-zero constant; (iv) f = αu, where α is constant (since M f dv f = 0). By (iii) and (iv), one has f f = α f u = αμ 2 f . We rewrite it by and then it infers that f must be zero on M since M is a closed manifold. Therefore, we complete the proof of Theorem 2.1 by the results of [6] and [11].
Proof of Theorem 2.2 In the following, we show an almost-Schur lemma under the assumption of m-Bakry-Émery Ricci tensor, which is similar to the work of [18]. Consider the nontrivial solution u : Additionally, the second Bianchi identity div Ric = 1 2 ∇R implies div Ric = n -2 2n ∇R, where (div Ric) j = ∇ i R ij and Ric = Ric -R n g. (2.13) Now we use Bochner's formula Here, we use Ric m f ≥ ( 1 m |∇f | 2 -(n -1)K)g. Therefore, by inequality (2.9) (but we replace (R f -R f ) with (R -R)), (2.14) gives (2.15) then (2.13) can be rewritten as That is, u is a weighted harmonic function with respect to weighted measure dv 2f m on M, it infers u = 0 on M. Thus, Theorem 2.2 follows by the results of [6] and [11].
By combining Theorem 2.2 and Theorem 2.1, we note the following property.
If α ≤ -1 μ , we rewrite (2.26) as (1 + αμ) |Hess u| 2 -( u) 2 + (n -1) + μ n ( u) 2 + μ n R + n(n -1)K u = 0 at p. (2.30) We note that at p, the n × n matrix Hess u must be semi-positive. Then |Hess u| 2 ≤ ( u) 2 at p, and the equality holds only if the rank of Hess u(p) is less than 2. From this inequality, each term on the left-hand side of (2.30) must be nonnegative. Therefore, u(p) = R(p) -R = 0, and then M is Einstein and has constant scalar curvature with respect to metric g.

Conclusion
This paper contributes two main points. One is that two types of almost-Schur inequalities on smooth metric measure spaces are established under m-Bakry-Émery Ricci conditions or ∞-Bakry-Émery Ricci conditions, which imply the results of Cheng [6] and De Lellis-Topping [11] whenever the weighted function f is constant. The other is that the equality of our inequality implies geometric qualities of manifold, because the equality holds if and only if the manifold is Einstein and has constant scalar curvature with respect to the background metric (see

Methods
In this paper, we show almost-Schur inequalities on smooth metric measure spaces. The key points in the proofs are ∇f ⊗ ∇u and Bochner's formula, then due to the Bianchi identity and the first positive eigenvalue of the weighted Laplacian, we establish the almost-Schur inequalities.