# Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Authors: Yawei Chu; Dehe Li; Jundong Zhou
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

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


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## Body

## 1. Introduction

Ann-dimensional n≥3 Riemannian manifold Mn,g is called a Ricci soliton if there exist a smooth vector field X and a constant λ∈ℝ on Mn such that(1)Rc+12LXg=λg,where Rc and LXg 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 Mn, the soliton is called a gradient Ricci soliton and (1) becomes(2)Rc+Hessf=λg.Note that whenX 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 ℝ2 or to a quotient of S2. The first rigidity theorem in dimension 3 was proved by Ivey [2] saying that a 3-dimensional compact gradient shrinker is a quotient of S3. 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 ℝ3 or to a quotient of ℝ×S2 or S3. Some relevant conclusions can be found in [4, 7, 8].Whenn≥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(3)Rm≤CR,which along with the fact Rm2=R∘m2+R2/6 implies(4)R∘m≤C2−16R2,and C≥6/6. 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]).
LetMn,g,f be an n-dimensional n≥3 complete gradient shrinker. If(5)R∘m≤1Cnλ−n−2nn−1R,then Mn,g is isometric to ℝn or a finite quotient of Sn.In this paper, we will restrict our attention to the rigidity of gradient shrinkers with pointwise pinched conditions associated withR∘m and the traceless Ricci tensor R∘c=Rc−R/ng. By establishing f-parabolic and algebraic curvature estimates, we prove two rigidity results for gradient shrinkers. More precisely, setting Cn=n−2/nn−1+n2−n−4/2n−2n−1nn+1, which is defined in Lemma 10, we have the following Theorem 2.Theorem 2.
Assume thatMn,g,f is a complete gradient shrinker of dimension n≥3. If(6)R∘m2≤1C2nλ−n−2nn−1R2+4R∘c2n,then Mn,g is isometric to ℝn or a finite quotient of Sn. Moreover, when the pinching condition in the right hand of (6) is weakened to(7)R∘m2<2n−1n−2λ−n−2nn−1R2+4R∘c2n,then Mn,g is Einstein.Remark 3.
When(8)R∘m≤1Cnλ−n−2nn−1R,by equation (6) and Theorem 2, we see that Mn,g is isometric to ℝn or a finite quotient of Sn. Therefore, Theorem 2 can be seen as a generalization of Theorem 1.Theorem 4.
LetMn,g,f be a complete gradient shrinker of dimension n≥3 with nonnegative Ricci curvature. If(9)R∘m≤1CnRnn−1+R∘c2R,then Mn,g is isometric to ℝn or a finite quotient of Sn.Remark 5.
As is shown in the proof, the condition of nonnegative Ricci curvature in Theorem4 can be relaxed to that Rc≤cR1+α for some constants c>0 and α≥0 satisfying cRα≥n−2/nn−1.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

LetMn,g be a connected Riemannian manifold of dimension n≥3. In local coordinates, denoting by Rijkl, Wijkl, and R∘jk=Rjk−R/ngjk 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]).(10)Rijkl=Wijkl+Rnn−1gilgjk−gikgjl+1n−2R∘ilgjk+R∘jkgil−R∘ikgjl−R∘jlgik.Correspondingly, the soliton equation (2) is rewritten as(11)Rij+∇i∇jf=λgij.Taking the trace in equation (11) gives(12)R+Δf=nλ.WritingR∘m=R∘ijkl=Rijkl−R/nn−1gilgjk−gikgjl and using the properties of Rm, one can easily derive the following equalities:(13)gilR∘ijkl=R∘jk,(14)R∘ijkl+R∘iljk+R∘iklj=0,(15)R∘ijkl=R∘klij=−R∘jikl=−R∘ijlk,(16)R∘m2=W2+4n−2R∘c2,where the norm of a 0,4-type tensor T is defined by(17)T2=Tijkl2=gimgjngksgltTijklTmnst=ΔTijklTijkl.Here and subsequently, the notationsu∗=ΔinfMnu as well as u∗=ΔsupMnu for a function u on Mn and Einstein summation convention are always adopted.Recall thef-Laplacian Δf, which is sometimes called the drifted Laplacian or Witten-Laplacian and is defined on a function u∈LiplocMn by(18)Δfu=Δu−g∇f,∇u=efdive−f∇u, in the weak sense, which is a self-adjoint operator on the space of square integrable functions on Mn,g,f with respect to weighted volume form e−fdVg. That is,(19)∫Mn∇φ·∇ψe−fdVg=−∫MnΔfφψe−fdVg,for any φ,ψ∈C0∞Mn, where dVg is the volume element induced by the metric g.First of all, we will compute thef-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 dimensionn≥3, we have(20)2R∘ijkl∇l∇pRijkp−12∇R∘m2·∇f=2λR∘m2−2R∘ijklR∘ijkpR∘lp−2RnR∘m2−4Rnn−1R∘c2.Proof.
For convenience, we set(21)A=2R∘ijkl∇l∇pRijkp−12∇R∘m2·∇f=2R∘ijkl∇l∇pRijkp−R∘ijkl∇pRijkl∇pf.☐On the one hand, by the second Bianchi identity, we get(22)−R∘ijkl∇pRijkl∇pf=−R∘ijkl∇lRijkp−∇kRijlp∇pf=−2R∘ijkl∇lRijkp∇pf.On the other hand, by the Ricci identity and the equation (11), we deduce that(23)2R∘ijkl∇l∇pRijkp=2R∘ijkl∇l∇iRjk−∇jRik=2R∘ijkl∇l∇j∇i∇kf−∇i∇j∇kf=2R∘ijkl∇lRijkp∇pf+2R∘ijklRijkpλglp−Rlp.Combining the factsRijkp=R∘ijkp+R/nn−1gipgjk−gikgjp and Rij=R∘ij+R/ngij with (23) yields(24)A=2λR∘m2−2R∘ijklR∘ijkp+Rnn−1gipgjk−gikgjpR∘lp+Rnglp.=2λR∘m2−2R∘ijklR∘ijkpR∘lp−2RnR∘m2−4Rnn−1R∘c2.Our next step is to compute the Laplacian ofR∘m2 for all Riemannian manifolds.Lemma 8.
LetMn,g be an n-dimensional n≥3 complete Riemannian manifold. Then,(25)12ΔR∘ijkl2=∇R∘m2−2R∘ijkl2R∘hilpR∘hjkp−12R∘ijhpR∘hpkl−R∘ijkhR∘lh+2R∘ijkl∇l∇pRijkp+2RnR∘m2−4Rnn−1R∘c2.Proof.
By the definitions ofR∘m, R∘m2, and (13), we have(26)12ΔR∘ijkl2=∇R∘m2+R∘ijkl∇p∇pR∘ijkl=∇R∘m2+R∘ijkl∇p∇pRijkl.☐Employing Bianchi’s second identity, we obtain(27)R∘ijkl∇p∇pRijkl=R∘ijkl∇p∇lRijkp−∇kRijlp=2R∘ijkl∇p∇lRijkp=2R∘ijkl∇p∇lR∘ijkp+2Rnn−1gipgjk=2R∘ijkl∇p∇lR∘ijkp+4R∘plnn−1∇p∇lR,which together with (26) implies(28)12ΔR∘ijkl2=∇R∘m2+2R∘ijkl∇p∇lR∘ijkp+4R∘plnn−1∇p∇lR.Making use of the Ricci identity and (13), we have(29)2R∘ijkl∇p∇lR∘ijkp=2R∘ijkl∇l∇pR∘ijkp−RplihR∘hjkp+RpljhR∘hikp−RplkhR∘ijhp−RplphR∘ijkh=2R∘ijkl∇l∇pR∘ijkp−R∘hilpR∘hjkp+R∘hjlpR∘hikp−R∘hklpR∘ijhp+2R∘ijklR∘ijkhR∘lh+Rnglh−2Rnn−1R∘ijklR∘hjkpghpgil−glhgip−R∘hikpghpgjl−glhgjp+R∘ijhpghpgkl−glhgkp=2R∘ijkl∇l∇pR∘ijkp−2R∘hilpR∘hjkp−R∘hklpR∘ijhp+2R∘ijklR∘ijkhR∘lh+2RnR∘m2−2Rnn−1R∘ijkl−R∘ljki−R∘ilkj−R∘ijlk+R∘jkgli−R∘ikglj.Combining (13) and (14) with (29), we get(30)2R∘ijkl∇p∇lR∘ijkl=−2R∘ijkl2R∘hilpR∘hjkp+R∘hklpR∘ijhp−R∘ijkhR∘lh+2R∘ijkl∇l∇pR∘ijkp+2RnR∘m2−4Rnn−1R∘c2,where(31)R∘ijklR∘hklpR∘ijhp=R∘ijklR∘ijhp−R∘hpkl−R∘hlpk=−12R∘ijklR∘ijhpR∘hpkl.Substituting (30) and (31) into (28), we obtain(32)12ΔR∘ijkl2=∇R∘m2−2R∘ijkl2R∘hilpR∘hjkp−12R∘ijhpR∘hpkl−R∘ijkhR∘lh+2R∘ijkl∇l∇pRijkp+2RnR∘m2−4Rnn−1R∘c2,where the formula(33)R∘ijkl∇l∇pR∘ijkp+2R∘plnn−1∇p∇lR=R∘ijkl∇l∇pRijkpis used in (32).By Lemmas7 and 8 and the fact that(34)12ΔfR∘ijkl2=12ΔR∘ijkl2−R∘ijkl∇pRijkl∇pf,we now arrive at the f-Laplacian formula of R∘m2 for all gradient Ricci solitons.Lemma 9.
LetMn,g,f be an n-dimensional n≥3 gradient Ricci soliton. Then,(35)12ΔfR∘ijkl2=∇R∘m2−2R∘ijkl2R∘hilpR∘hjkp−12R∘ijhpR∘hpkl+2λR∘m2−8Rnn−1R∘c2.Consequently, we conclude from (35) and (16) that(36)12ΔfR∘ijkl2=∇R∘m2−2R∘ijkl2R∘hilpR∘hjkp−12R∘ijhpR∘hpkl+2λR∘m2−2n−2Rnn−1R∘m2+2n−2Rnn−1W2,for any gradient Ricci soliton.Utilizing the inequalities proved by Li and Zhao [15] and Huisken [16] (see also [17]), we have(37)2R∘ijklR∘hilpR∘hjkp≤n−2nn−1R∘m3,(38)R∘ijklR∘ijhpR∘hpkl≤n2−n−4n−2n−1nn+1R∘m3.Combining (37) and (38) with (36) givesLemma 10.
LetMn,g,f be an n-dimensional n≥3 gradient Ricci soliton. Then,(39)12ΔfR∘m2≥∇R∘m2−2n−2nn−1+n2−n−42n−2n−1nn+1R∘m3+2λR∘m2−2n−2Rnn−1R∘m2+2n−2Rnn−1W2=Δ∇R∘m2−2CnR∘m3+2λ−n−2Rnn−1R∘m2+2n−2Rnn−1W2.Remark 11.
Inequality (39) can also be derived by setting X=∇f in Lemma 2.5 of [12]; here, we give its proof for the sake of completeness.Correspondingly, thef-Laplacian of R∘c2 is (see e.g. Lemma 2.1 of [18])Lemma 12.
LetMn,g,f be a gradient Ricci soliton of dimension n≥3. Then,(40)12ΔfR∘c2=∇R∘c2+2λ−n−2Rnn−1R∘c2+4n−2R∘ijR∘jkR∘ki−2WijklR∘ilR∘jk.Employ the following curvature inequality.Lemma 13. (Proposition 2.1 of [19]).
LetMn,g be an n-dimensional n≥3 Riemannian manifold. Then,(41)−WijklR∘ilR∘jk+2n−2R∘ijR∘ipR∘pj≤n−22n−1W2+8nn−2R∘c212R∘c2.We deduce from Lemma12 the following.Lemma 14.
LetMn,g be an n-dimensional n≥3 Riemannian manifold. Then,(42)12ΔfR∘c2≥∇R∘c2+2λ−n−2Rnn−1−n−22n−1W2+8nn−2R∘c212R∘c2.Since the scalar curvature of nonflat Ricci shrinker is positive, by Proposition 2.7 of [12], we get the following curvature inequality.Lemma 15.
LetMn,g,f be an n-dimensional n≥3 complete nonflat gradient shrinker. Then,(43)Δf−2∇logRR∘m2R2≥2R3R∇R∘m−R∘m∇RR2+4n−2W2nn−1R+4R∘m2R3R2nn−1+R∘c2−CnRR∘m.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]).
LetMn,g,f be a complete gradient nonflat shrinker of dimension n≥3. Then,(44)Δf−2∇logRR∘c2R2≥2R4R∇jRik−∇jRRik2+4R3R∘c2R∘c−Rnn−12−RWijklR∘ilR∘jk.Lemma 17 (Theorem 3 of [20]).
LetMn,g,f be a complete gradient shrinker of dimension n≥3. Then, 0≤R∗≤nλ. Moreover, R∗<nλ unless M,g is Einstein and the soliton is trivial, and R∗>0 unless R≡0 and M,g is isometric to ℝn.Lemma 18 (Theorem 22 of [20]).
Any complete gradient shrinkerMn,g,f is f-parabolic, namely, every solution of Δfu≥0 satisfying u∗<+∞ must be a constant.Lemma 19 (Lemma 4.2 of [8]).
Assume thatMn,g is an n-dimensional manifold with finite w-volume, i.e., ∫Me−wdVg<+∞. If a smooth function u∈L2e−wdVg is bounded below such that Δwu≥0, then u 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:(45)W2+8nn−2R∘c2≤1C2nλ−n−2nn−1R2,(46)W2+8nn−2R∘c2<2n−1n−2λ−n−2nn−1R2.By Lemma14 and (46), we have(47)12ΔfR∘c2≥∇R∘c2+2λ−n−2Rnn−1−n−22n−1W2+8nn−2R∘c21/2R∘c2≥2λ−n−2Rnn−1−n−22n−1W2+8nn−2R∘c21/2R∘c2≥0, which along with Lemma 18 and (46) yields R∘c=0 and therefore Mn,g is Einstein.On the other hand, if (45) holds, it is easily seen that (46) also holds. Indeed, when n=3, clearly 1/C23=2/3<4. When n≥4, we see from the fact(48)Cn=n−2nn−1+n2−n−42n−2n−1nn+1>n−2nn−1that(49)1C2n<nn−1n−22≤2n−1n−2.It follows from Lemma18 and (45) that R∘c=0, which together with (16) implies(50)R∘m2=W2≤1C2nλ−n−2nn−1R2.By Lemma10 and (50) we know that(51)12ΔfR∘m2≥∇R∘m2+2λ−n−2Rnn−1−CnR∘mR∘m2+2n−2Rnn−1W2≥2λ−n−2Rnn−1−CnR∘mR∘m2≥0,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 R∘m is a constant and therefore all equalities in (51) hold.If there existsx0∈Mn such Rx0=0, then, we see from Lemma 17 that Mn is isometric to ℝn.Otherwise, the factsR>0, R∘c=0 and the equalities of (51) imply that R∘m=W=0. Hence, we know that Mn,g has constant sectional curvature when R>0; it follows from the Myers theorem and the condition R>0 that Mn,g is compact and therefore is a finite quotient ofSn.Proof of Theorem 4.
It is well known thatR≥0 for shrinking solitons. When R achieves its infimum 0, Lemma 17 says that Mn,g,f is flat and therefore is isometric to ℝn.In the rest of the proofs of Theorem4, we assume that R>0. By (43) and (9), we see that(52)Δf−2∇logRR∘m2R2≥2R3R∇R∘m−R∘m∇RR2+4n−2W2nn−1R+4R∘m2R3R2nn−1+R∘c2−CnRR∘m≥4R∘m2R3R2nn−1+R∘c2−CnRR∘m≥0.Setw=f−∇logR2 and u=R∘m2/R2. In order to apply Lemma 19 to (52), we need to verify(53)∫Mne−wdVg=∫MnR2e−fdVg<+∞,(54)∫MnR∘m4R4e−wdVg=∫MnR∘m4R2e−fdVg<+∞.In fact, (53) follows from the result that R∈Lpe−fdVg for 1≤p<+∞ (see e.g., [10]) for all gradient shrinkers.Under the assumption thatMn,g,f has nonnegative Ricci curvature and (9), we get ∣Rc∣≤R. Thus,(55)R∘m≤1CnR−n−2Rnn−1=n2−2n+2nn−1CnR,R∘m4R2≤n2−2n+2nn−1Cn4R2∈Le−fdVg.Furthermore, as observed in Remark5, if we relax the condition of negative Ricci curvature to that Rc≤cR1+α for some constants c>0 and α≥0 satisfying cRα≥n−2/nn−1, then(56)R∘m≤RCnc2R2α−n−2nn−1.It follows from the resultR∈Lpe−fdVg for 1≤p<+∞ that(57)R∘m4R2∈Le−fdVg.These together with (52) and Lemma 19 yield R∘m=0 or W=0 and R∘m=1/CnR/nn−1+R∘c2/R.What we need to prove now is thatR∘c=0 in the latter case. In fact, by (16), the facts W=0 and R∘m=1/CnR/nn−1+R∘c2/R, we derive(58)R∘c=n−22R∘m=n−22CnRnn−1+R∘c2R.It is easy to check from the definition ofCn that the different two solutions of equation (58) satisfy(59)R∘c1=CnRn−2+RC2nn−2−1nn−1>n−2nn−1R,(60)R∘c2=1/nn−1R2CnR/n−2+RC2n/n−2−1/nn−1<Rn−2n−1n.Combining (59) and (60) and the fact W=0 with Lemma 16 gives(61)Δf−2∇logRR∘c2R2≥2R4R∇jRik−∇jRRik2+4R∘c2R3R∘c−Rnn−12≥4R∘c2R3R∘c−Rnn−12≥0.By a similar argument, we conclude from Lemma19 and the assumption on the Ricci curvature that R∘c2/R is a constant and R∘c=0 since R∘c1,2≠R/nn−1. This concludes the proof of Theorem 4.



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