Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Abstract

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

1. Introduction

An -dimensional Riemannian manifold is called a Ricci soliton if there exist a smooth vector field and a constant on such thatwhere and denote the Ricci tensor and the Lie derivative of in the direction of , respectively, and is sometimes called the soliton constant. The soliton is shrinking, steady, or expanding if , , or , respectively. When is a gradient of a smooth function on , the soliton is called a gradient Ricci soliton and (1) becomes

Note that when or 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 or 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 or to a quotient of . The first rigidity theorem in dimension 3 was proved by Ivey [2] saying that a -dimensional compact gradient shrinker is a quotient of . 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 -dimensional gradient shrinker must be isometric to or to a quotient of or . Some relevant conclusions can be found in [4, 7, 8].

When , 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 -dimensional gradient shrinker and proved that there exists a constant for -dimensional gradient shrinkers with bounded scalar curvature so thatwhich along with the fact impliesand . Here, is the trace-free curvature tensor.

In [12], the authors established the following rigidity theorem under pointwise pinching condition of:

Theorem 1 (Theorem 1.1 in [12]). Let be an -dimensional complete gradient shrinker. Ifthen is isometric to or a finite quotient of .

In this paper, we will restrict our attention to the rigidity of gradient shrinkers with pointwise pinched conditions associated with and the traceless Ricci tensor . By establishing -parabolic and algebraic curvature estimates, we prove two rigidity results for gradient shrinkers. More precisely, setting , which is defined in Lemma 10, we have the following Theorem 2.

Theorem 2. Assume that is a complete gradient shrinker of dimension . Ifthen is isometric to or a finite quotient of . Moreover, when the pinching condition in the right hand of (6) is weakened tothen is Einstein.

Remark 3. When by equation (6) and Theorem 2, we see that is isometric to or a finite quotient of . Therefore, Theorem 2 can be seen as a generalization of Theorem 1.

Theorem 4. Let be a complete gradient shrinker of dimension with nonnegative Ricci curvature. Ifthen is isometric to or a finite quotient of .

Remark 5. As is shown in the proof, the condition of nonnegative Ricci curvature in Theorem 4 can be relaxed to that for some constants and satisfying .

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.

2. Preliminaries of Curvature Estimates

Let be a connected Riemannian manifold of dimension . In local coordinates, denoting by , , and the components of the curvature tensor , the Weyl tensor , and the traceless Ricci tensor , respectively, we have the well-known orthogonal decomposition of (see e.g., [14]).

Correspondingly, the soliton equation (2) is rewritten as

Taking the trace in equation (11) gives

Writing and using the properties of , one can easily derive the following equalities:where the norm of a -type tensor is defined by

Here and subsequently, the notations as well as for a function on and Einstein summation convention are always adopted.

Recall the -Laplacian , which is sometimes called the drifted Laplacian or Witten-Laplacian and is defined on a function by in the weak sense, which is a self-adjoint operator on the space of square integrable functions on with respect to weighted volume form . That is,for any , where is the volume element induced by the metric .

First of all, we will compute the -Laplacian of the norm square of , by which we will establish the key estimate for any gradient Ricci soliton of dimension in Lemma 10. We start from Lemma 7.

Lemma 7. For any gradient Ricci soliton of dimension , we have

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 and with (23) yields

Our next step is to compute the Laplacian of for all Riemannian manifolds.

Lemma 8. Let be an -dimensional complete Riemannian manifold. Then,

Proof. By the definitions of , , and (13), we have☐

Employing Bianchi’s second identity, we obtainwhich together with (26) implies

Making use of the Ricci identity and (13), we have

Combining (13) and (14) with (29), we getwhere

Substituting (30) and (31) into (28), we obtainwhere the formulais used in (32).

By Lemmas 7 and 8 and the fact thatwe now arrive at the -Laplacian formula of for all gradient Ricci solitons.

Lemma 9. Let be an -dimensional gradient Ricci soliton. Then,

Consequently, we conclude from (35) and (16) thatfor any gradient Ricci soliton.

Utilizing the inequalities proved by Li and Zhao [15] and Huisken [16] (see also [17]), we have

Combining (37) and (38) with (36) gives

Lemma 10. Let be an -dimensional gradient Ricci soliton. Then,

Remark 11. Inequality (39) can also be derived by setting in Lemma 2.5 of [12]; here, we give its proof for the sake of completeness.

Correspondingly, the -Laplacian of is (see e.g. Lemma 2.1 of [18])

Lemma 12. Let be a gradient Ricci soliton of dimension . Then,

Employ the following curvature inequality.

Lemma 13. (Proposition 2.1 of [19]). Let be an -dimensional Riemannian manifold. Then,

We deduce from Lemma 12 the following.

Lemma 14. Let be an -dimensional 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 15. Let be an -dimensional complete nonflat gradient shrinker. Then,

For the sake of the proofs of our main theorems, we recall the following results due to Catino [10], Pigola et al. [20], and Petersen and Wylie [8] (see also [4]).

Lemma 16 (Proposition 1 of [10]). Let be a complete gradient nonflat shrinker of dimension . Then,

Lemma 17 (Theorem 3 of [20]). Let be a complete gradient shrinker of dimension . Then, Moreover, unless is Einstein and the soliton is trivial, and unless and is isometric to .

Lemma 18 (Theorem 22 of [20]). Any complete gradient shrinker is -parabolic, namely, every solution of satisfying must be a constant.

Lemma 19 (Lemma 4.2 of [8]). Assume that is an -dimensional manifold with finite -volume, i.e., . If a smooth function is bounded below such that , then is a constant.

3. 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:

By Lemma 14 and (46), we have which along with Lemma 18 and (46) yields and therefore is Einstein.

On the other hand, if (45) holds, it is easily seen that (46) also holds. Indeed, when , clearly . When , we see from the factthat

It follows from Lemma 18 and (45) that , which together with (16) implies

By Lemma 10 and (50) we know thatwhere the fact that 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 is a constant and therefore all equalities in (51) hold.

If there exists such , then, we see from Lemma 17 that is isometric to .

Otherwise, the facts , and the equalities of (51) imply that . Hence, we know that has constant sectional curvature when ; it follows from the Myers theorem and the condition that is compact and therefore is a finite quotient of.

Proof of Theorem 4. It is well known that for shrinking solitons. When achieves its infimum , Lemma 17 says that is flat and therefore is isometric to .

In the rest of the proofs of Theorem 4, we assume that By (43) and (9), we see that

Set and . In order to apply Lemma 19 to (52), we need to verify

In fact, (53) follows from the result that for (see e.g., [10]) for all gradient shrinkers.

Under the assumption that has nonnegative Ricci curvature and (9), we get . Thus,

Furthermore, as observed in Remark 5, if we relax the condition of negative Ricci curvature to that for some constants and satisfying , then

It follows from the result for that

These together with (52) and Lemma 19 yield or and .

What we need to prove now is that in the latter case. In fact, by (16), the facts and , we derive

It is easy to check from the definition of that the different two solutions of equation (58) satisfy

Combining (59) and (60) and the fact with Lemma 16 gives

By a similar argument, we conclude from Lemma 19 and the assumption on the Ricci curvature that is a constant and since . 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.

Acknowledgments

The first author would like to express his sincere gratitude to Prof. Xi Zhang and University of Science and Technology of China for their support. The research was partially supported by the project “Analysis and Geometry on Bundle” of the Ministry of Science and Technology of the People’s Republic of China (No. SQ2020YFA070080) and NSF of China (Nos. 11625106, 11801535, 11721101, and 11801011). The authors were partially supported by the NSF of Anhui Provincial Education Department (Nos. KJ2018A0330, gxgnfx2018017, 2019mooc205, and 2019MSZGS01).