Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Let ð M n , g , f Þ be a complete gradient shrinking Ricci soliton of dimension n ≥ 3 . In this paper, we study the rigidity of ð M n , g , f Þ with pointwise pinching curvature and obtain some rigidity results. In particular, we prove that every n -dimensional gradient shrinking Ricci soliton ð M n , g , f Þ is isometric to ℝ n or a ﬁ nite quotient of S n under some pointwise pinching curvature condition. The arguments mainly rely on algebraic curvature estimates and several analysis tools on ð M n , g , f Þ , such as the property of f -parabolic and a Liouville type theorem.


Introduction
An n-dimensional ðn ≥ 3Þ Riemannian manifold ðM n , gÞ is called a Ricci soliton if there exist a smooth vector field X and a constant λ ∈ ℝ on M n such that where Rc and L X g denote the Ricci tensor and the Lie derivative of g in the direction of X, respectively, and λ is sometimes called the soliton constant. The soliton is shrinking, steady, or expanding if λ > 0, λ = 0, or λ < 0, respectively. When X is a gradient of a smooth function f on M n , the soliton is called a gradient Ricci soliton and (1) becomes Rc + Hess f = λg: ð2Þ Note that when X or ∇f is a Killing vector field, equations (1) and (2) reduce to the Einstein equation. Thus, Ricci solitons are natural generalizations of Einstein maifolds. In particular, when X = 0 or f is a constant, the soliton is trivial.
In recent decades, increasing investigations have been done to the rigidity of gradient shrinking Ricci solitons (gradient shrinker for short). In dimension 2, Hamilton [1] showed that a gradient shrinker is isometric to ℝ 2 or to a quotient of S 2 . The first rigidity theorem in dimension 3 was proved by Ivey [2] saying that a 3-dimensional compact gradient shrinker is a quotient of S 3 . In the noncompact case, the revelent rigidity result was showed by Perelman [3] with noncollapsing assumption, which was removed by Naber [4] later. Adopting different arguments, Ni and Wallach [5] and Cao et al. [6] obtained the full classification; they proved that any 3-dimensional gradient shrinker must be isometric to ℝ 3 or to a quotient of ℝ × S 2 or S 3 . Some relevant conclusions can be found in [4,7,8].
When n ≥ 4, under the assumption of nonnegative curvature operator or vanishing Weyl tensor, Naber [4], Ni and Wallach [5], Petersen and Wylie [8], and Zhang [9] proved corresponding rigidity results on gradient shrinkers, which were improved by Catino [10] using a general pointwise pinching condition on the Weyl tensor.
On the other hand, Munteanu and Wang [11] investigated the curvature behavior of 4-dimensional gradient shrinker and proved that there exists a constant C > 0 for 4-dimensional gradient shrinkers with bounded scalar curvature R so that which along with the fact jRmj 2 = j R ∘ mj 2 + ðR 2 /6Þ implies and C ≥ ð ffiffi ffi 6 p /6Þ. Here, R ∘ m is the trace-free curvature tensor.
In [12], the authors established the following rigidity theorem under pointwise pinching condition ofR ∘ m: Theorem 1 (Theorem 1.1 in [12]). Let ðM n , g, f Þ be an n -dimensional ðn ≥ 3Þ complete gradient shrinker. If then ðM n , gÞ is isometric to ℝ n or a finite quotient of S n .
In this paper, we will restrict our attention to the rigidity of gradient shrinkers with pointwise pinched conditions associated with R ∘ m and the traceless Ricci tensor R ∘ c = Rc − ðR/nÞg. By establishing f -parabolic and algebraic curvature estimates, we prove two rigidity results for gradient shrinkers. More precisely, setting ÞÞ, which is defined in Lemma 10, we have the following Theorem 2.
Theorem 2. Assume that ðM n , g, f Þ is a complete gradient shrinker of dimension n ≥ 3. If then ðM n , gÞ is isometric to ℝ n or a finite quotient of S n . Moreover, when the pinching condition in the right hand of (6) is weakened to then ðM n , gÞ is Einstein.
by equation (6) and Theorem 2, we see that ðM n , gÞ is isometric to ℝ n or a finite quotient of S n . Therefore, Theorem 2 can be seen as a generalization of Theorem 1.

Theorem 4.
Let ðM n , g, f Þ be a complete gradient shrinker of dimension n ≥ 3 with nonnegative Ricci curvature. If then ðM n , gÞ is isometric to ℝ n or a finite quotient of S n .

Remark 5.
As is shown in the proof, the condition of nonnegative Ricci curvature in Theorem 4 can be relaxed to that jRcj ≤ cR 1+α for some constants c > 0 and α ≥ 0 satisfying cR α ≥ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðn − 2Þ/ðnðn − 1ÞÞ p .
Remark 6. Since any three-dimensional gradient shrinker must have nonnegative sectional curvature (cf. Corollary 2.4 of [13]), we see that the condition on Ricci curvature in Theorem 4 is not needed.

Preliminaries of Curvature Estimates
Let ðM n , gÞ be a connected Riemannian manifold of dimension n ≥ 3. In local coordinates, denoting by R ijkl , W ijkl , and R ∘ jk = R jk − ðR/nÞg jk the components of the curvature tensor Rm, the Weyl tensor W, and the traceless Ricci tensor R ∘ c, respectively, we have the well-known orthogonal decomposition of Rm (see e.g., [14]).
Correspondingly, the soliton equation (2) is rewritten as Taking the trace in equation (11) gives Writing R ∘ m = fR ∘ ijkl g = fR ijkl − ðR/nðn − 1ÞÞðg il g jk − g ik g jl Þg and using the properties of Rm, one can easily derive the following equalities: 2 Advances in Mathematical Physics where the norm of a ð0, 4Þ-type tensor T is defined by Here and subsequently, the notations u * = Δ inf M n u as well as u * = Δ sup M n u for a function u on M n and Einstein summation convention are always adopted.
Recall the f -Laplacian Δ f , which is sometimes called the drifted Laplacian or Witten-Laplacian and is defined on a function u ∈ Lip loc ðM n Þ by in the weak sense, which is a self-adjoint operator on the space of square integrable functions on ðM n , g, f Þ with respect to weighted volume form e −f dV g . That is, for any φ, ψ ∈ C ∞ 0 ðM n Þ, where dV g is the volume element induced by the metric g.
First of all, we will compute the f -Laplacian of the norm square of R ∘ m, by which we will establish the key estimate for any gradient Ricci soliton of dimension n ≥ 3 in Lemma 10. We start from Lemma 7.

Lemma 7. For any gradient Ricci soliton of dimension
Proof. For convenience, we set On the one hand, by the second Bianchi identity, we get On the other hand, by the Ricci identity and the equation (11), we deduce that Combining the facts Our next step is to compute the Laplacian of j R ∘ mj 2 for all Riemannian manifolds. Lemma 8. Let ðM n , gÞ be an n-dimensional ðn ≥ 3Þ complete Riemannian manifold. Then, Proof. By the definitions of R ∘ m, j R ∘ mj 2 , and (13), we have ☐ Employing Bianchi's second identity, we obtain which together with (26) implies

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Making use of the Ricci identity and (13), we have Combining (13) and (14) with (29), we get where Substituting (30) and (31) into (28), we obtain where the formula is used in (32). By Lemmas 7 and 8 and the fact that we now arrive at the f -Laplacian formula of j R ∘ mj 2 for all gradient Ricci solitons.

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Correspondingly, the f -Laplacian of j R ∘ cj 2 is (see e.g. Lemma 2.1 of [18]) Lemma 12. Let ðM n , g, f Þ be a gradient Ricci soliton of dimension n ≥ 3. Then, Employ the following curvature inequality.

Lemma 14.
Let ðM n , gÞ be an n -dimensional ðn ≥ 3Þ Riemannian manifold. Then, Since the scalar curvature of nonflat Ricci shrinker is positive, by Proposition 2.7 of [12], we get the following curvature inequality.

Lemma 19 (Lemma 4.2 of [8])
. Assume that ðM n , gÞ is an n-dimensional manifold with finite w -volume, i.e., Ð M e −w d V g < +∞. If a smooth function u ∈ L 2 ðe −w dV g Þ is bounded below such that Δ w u ≥ 0, then u is a constant.

Proofs of Main Theorems
We are now in a position to give the proofs of our main theorems.
Proof of Theorem 2. Using (16), we see that pinching conditions (6) and (7) in Theorem 2 are equivalent to the following inequality, respectively:

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By Lemma 14 and (46), we have which along with Lemma 18 and (46) yields j R ∘ cj = 0 and therefore ðM n , gÞ is Einstein.
On the other hand, if (45) holds, it is easily seen that (46) also holds. Indeed, when n = 3, clearly 1/C 2 ð3Þ = 2/3 < 4. When n ≥ 4, we see from the fact It follows from Lemma 18 and (45) that j R ∘ cj = 0, which together with (16) implies By Lemma 10 and (50) we know that where the fact that R ≥ 0 for shrinking solitons (see Lemma 17 or Corollary 2.5 of [13]) is used in the second inequality in (51). It follows from Lemma 18 and (50) that j R ∘ mj is a constant and therefore all equalities in (51) hold. If there exists x 0 ∈ M n such Rðx 0 Þ = 0, then, we see from Lemma 17 that M n is isometric to ℝ n .
Otherwise, the facts R > 0, j R ∘ cj = 0 and the equalities of (51) imply that R ∘ m = W = 0. Hence, we know that ðM n , gÞ has constant sectional curvature when R > 0; it follows from the Myers theorem and the condition R > 0 that ðM n , gÞ is compact and therefore is a finite quotient ofS n .
Proof of Theorem 4. It is well known that R ≥ 0 for shrinking solitons. When R achieves its infimum 0, Lemma 17 says that ðM n , g, f Þ is flat and therefore is isometric to ℝ n .
What we need to prove now is that R ∘ c = 0 in the latter case. In fact, by (16), the facts W = 0 and j R ∘ mj = 1/CðnÞððR/ðnðn − 1ÞÞÞ + ðj R ∘ cj 2 /RÞÞ, we derive It is easy to check from the definition of CðnÞ that the different two solutions of equation (58) satisfy Combining (59) and (60) and the fact W = 0 with Lemma 16 gives By a similar argument, we conclude from Lemma 19 and the assumption on the Ricci curvature that j R ∘ cj 2 /R is a constant and R ∘ c = 0 since j R ∘ cj 1,2 ≠ R/ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn − 1Þ p . This concludes the proof of Theorem 4.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.