Geometric inequalities and rigidity of gradient shrinking Ricci solitons
aa r X i v : . [ m a t h . DG ] S e p GEOMETRIC INEQUALITIES AND RIGIDITY OF GRADIENTSHRINKING RICCI SOLITONS
JIA-YONG WU
Abstract.
In this paper we prove that the Sobolev inequality, the logarithmic Sobolevinequality, the Schr¨odinger heat kernel upper bound, the Faber-Krahn inequality, the Nashinequality and the Rozenblum-Cwikel-Lieb inequality all equivalently exist on completegradient shrinking Ricci solitons. We also obtain some integral gap theorems for compactshrinking Ricci solitons. Introduction
In this paper we will investigate the equivalence of various geometric inequalities on gradi-ent shrinking Ricci solitons. As applications, We apply the Sobolev inequality to give someintegral gap theorems for compact shrinking Ricci solitons. Recall that an n -dimensionalRiemannian manifold ( M, g ) is called a gradient shrinking Ricci soliton or shrinkers (see[28]) if there exists a smooth function f on M such that the Ricci curvature Ric and theHessian of f satisfy Ric + Hess f = λg (1.1)for some positive number λ . Function f is often called a potential of the shrinker. Forsimplicity, we often normalize λ = by scaling the metric g so thatRic + Hess f = 12 g. (1.2)According to the work of [28, 9], without loss of generality, adding f by a constant ifnecessary, we can assume that equation (1.2) simultaneously satisfies(1.3) R + |∇ f | = f and (4 π ) − n Z M e − f dv = e µ , where R is the scalar curvature of ( M, g ) and µ = µ ( g,
1) is the entropy functional ofPerelman [46]; see also the detailed explanation in [37] or [55, 56]. For a compact shrinker, µ has a lower bound; but for the non-compact case, we generally need to assume µ > −∞ so that our discussion makes sense. In particular, for the Euclidean space, we have µ = 0.In [37], Li, Li and Wang proved that e µ is nearly equivalent to V ( p , B ( p ,
1) centered at point p ∈ M and radius 1. Here p ∈ M is a point where f attains its infimum, which always exists on shrinkers but possibly is not unqiue; see [29]. Date : September 29, 2020.2010
Mathematics Subject Classification.
Primary 53C25, 58J35; Secondary 53C20, 53C24.
Key words and phrases. shrinking Ricci soliton, Schr¨odinger operator, Sobolev inequality, logarithmicSobolev inequality, heat kernel, Faber-Krahn inequality, Nash inequality, Rozenblum-Cwikel-Lieb inequality,eigenvalue, half Weyl tensor, rigidity.
In this paper we always let the triple ( M, g, f ) denote the shrinking gradient Ricci soliton(or shrinker) with (1.2) and (1.3) satisfying µ > −∞ . Shrinkers are natural extension of Einstein manifolds and can be regarded as the criticalpoint of Perelman’s W -functional [46]. In addition, shrinkers are self-similar solutions tothe Ricci flow and naturally rise as singularity analysis of the Ricci flow [28]. For example,Enders, M¨uller and Topping [20] proved that the proper rescaling limits of a type-I singu-larity point always converge to non-trivial shrinkers. At present, one of main issue in theRicci flow theory is the understanding on the geometry and classification of shrinkers. Fordimensions 2 and 3, the classification is complete by the works of [27], [35], [46], [44] and[5]. For dimension 4 and higher, the classification remains open, though much progress hasbeen made. The interested reader can refer to [4] for an excellent survey.In recent years, many geometric and analytic results about shrinkers have been inves-tigated. Wylie [61] proved that any complete shrinker has finite fundamental group (thecompact case due to Derdzi´nski [18]). Chen [14] showed that the scalar curvature R ≥ > M, g, f ) is the Gaussian shrinker;Chow, Lu and Yang [3] showed that the scalar curvature of non-trivial shrinkers has at leastquadratic decay of distance function. Chen and Zhou [7] showed that the potential function f is uniformly equivalent to the distance squared; they [7] also showed that all shrinkers haveat most Euclidean volume growth by combining an observation of Munteanu [41]. Later,Munteanu and Wang [42] proved that shrinkers have at least linear volume growth. Thesevolume growth properties are similar to manifolds with nonnegative Ricci curvature.On the other hand, Haslhofer and M¨uller [29, 30] proved a Cheeger-Gromov compactnesstheorem of shrinker. Huang [33] proved an ǫ -regularity theorem for 4-dimensional shrinkers,which was later improved by Ge and Jiang [22]. Their result gives an answer to Cheeger-Tian’s question [13]. In [53], the author applied gradient estimate technique to prove aLiouvlle type theorem for ancient solutions to the weighted heat equation on shrinkers. In[57, 58], P. Wu and the author applied weighted heat kernel upper estimates to give a sharpweighted L -Liouville theorem for weighted subharmonic functions on shrinkers.By analyzing the Perelman’s functional under the Ricci flow, Li and Wang [39] obtained asharp logarithmic Sobolev inequality on complete (possible non-compact) shrinkers, whichsays that for any τ > Z M ϕ ln ϕ dv ≤ τ Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv − (cid:16) µ + n + n πτ ) (cid:17) . for any compactly supported locally Lipschitz function ϕ with R M ϕ dv = 1. The equalitycase could be attained when τ = 1 (see Carrillo and Ni [9]). This sharp inequality is usefulfor understanding shrinkers. Indeed the author [56] was able to apply (1.4) to give sharpupper diameter bounds of compact shrinkers in terms of the integral of scalar curvature.The author [55] also used (1.4) to study the Schrodinger heat kernel on shrinkers. Here thedefinition of the Schr¨odinger heat kernel is similar to the classical heat kernel. That is, foreach y ∈ M , we say that H R ( x, y, t ) is called the Schrodinger heat kernel if H R ( x, y, t ) = u ( x, t ) is a minimal positive smooth solution of the Schr¨odinger heat equation − ∆ u + R4 u + ∂ t u = 0 EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 3 satisfying lim t → u ( x, t ) = δ y ( x ), where δ y ( x ) is the delta function defined as Z M φ ( x ) δ y ( x ) dv = φ ( y )for any φ ∈ C ∞ ( M ). In general, the Schr¨odinger heat kernel always exists on compactshrinkers. For non-compact shrinkers, we do not know if it still exists without any assump-tion but it indeed exists when the scalar curvature of ( M, g, f ) is bounded. The Schr¨odingerheat kernel of shrinkers shares many kernel properties of the classical Laplacian heat kernelon manifolds; see [55]. In this paper, we always assume that the Schr¨odinger heat kernelexists on n -dimensional complete shrinker ( M, g, f ).In [55] the author applied (1.4) to prove that(1.5) H R ( x, y, t ) ≤ e − µ (4 πt ) n for all x, y ∈ M and t >
0. We remark that this type upper bound of the conjugate heatkernel under the Ricci flow was obtained by Li and Wang [39]. The author also obtained itsGaussian type upper bounds by the iteration argument. That is, for any α >
4, the authorshowed that there exists a constant A = A ( n, α ) depending on n and α such that(1.6) H R ( x, y, t ) ≤ Ae − µ (4 πt ) n exp (cid:18) − d ( x, y ) αt (cid:19) for all x, y ∈ M and t >
0, where d ( x, y ) denotes the geodesic distance between x and y . Con-sidering the classical Laplace heat kernel of Euclidean space, estimate (1.6) is obvious sharp.Moreover the heat kernel estimate is useful for analyzing eigenvalues of the Schr¨odinger oper-ator. Indeed the author [55] used the upper bounds to get lower bounds of their eigenvalues.Namely, for any open relatively compact set Ω ⊂ M , let 0 < λ (Ω) ≤ λ (Ω) ≤ . . . be theDirichlet eigenvalues of the Schr¨odinger operator in Ω. Then we have(1.7) λ k (Ω) ≥ nπe (cid:18) k e µ V (Ω) (cid:19) /n , k ≥ , where V (Ω) is the volume of Ω. Recall that the classical Weyl’s asymptotic formula of the k -th Dirichlet eigenvalue of Laplacian in open relatively compact set Ω ⊂ R n states that λ k (Ω) ∼ c ( n ) (cid:18) kV (Ω) (cid:19) /n , k → ∞ , which indicates that (1.7) is sharp for the exponent 2 /n . We remark that by the Rozenblum-Cwikel-Lieb inequality [49, 16, 40] (see also estimate (3.1) in Section 3), we easily get thateigenvalues of the Schr¨odinger operator − ∆ + R4 on non-trivial shrinkers are all positive.Besides of the above results, combining (1.4) and the Markov semigroup technique ofDavies [17], Li and Wang [39] proved a local Sobolev inequality of shrinkers. Namely, foreach compactly supported locally Lipschitz function ϕ in M ,(1.8) (cid:18)Z M ϕ nn − dv (cid:19) n − n ≤ C ( n ) e − µn Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv JIA-YONG WU for some constant C ( n ) depending only on n . Here e µ can be viewed as the volume ofunit geodesic ball and hence this Sobolev inequality is very similar to the classical Sobolevinequality on manifolds, which plays an important role in some PDE ways. For example, P.Wu and the author [59] applied (1.8) to study the dimensional estimates for the spaces ofharmonic functions and Schr¨odinger functions with polynomial growth. In [54], the authorused (1.8) to derive a mean value type inequality and further study the analyticity in timefor solutions of the heat equation on shrinkers.In this paper we continue to study geometric inequalities and their relations on shrinkers.We first show that the above geometric inequalities, and the Nash inequality, the Rozenblum-Cwikel-Lieb inequality all equivalently exist on shrinkers, which may be regarded as naturalgeneralizations of the case of manifolds with nonnegative Ricci curvature [23, 50, 62]. Theorem 1.1.
Let ( M, g, f ) be an n -dimensional complete (compact or noncompact) shrinker.The following six properties are equivalent up to constants. (I) The Sobolev inequality (1.8) holds. (II)
The logarithmic Sobolev inequality (1.4) holds. (III)
The Schr¨odinger heat kernel upper bound (1.6) holds. (IV)
The Faber-Krahn inequality holds. That is, for all open relatively compact set Ω ⊂ M with smooth boundary, λ (Ω) ≥ nπe (cid:18) e µ V (Ω) (cid:19) n , where λ (Ω) is the lowest Dirichlet eigenvalue of the Schr¨odinger operator in Ω . (V) The Nash inequality holds. That is, there exists a constant c ( n ) depending on n suchthat (1.9) k ϕ k n ≤ c ( n ) e − µn k ϕ k n Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv for any compactly supported locally Lipschitz function ϕ in M . (VI) The Rozenblum-Cwikel-Lieb inequality holds. That is, there exists a constant c ( n ) depending on n such that (1.10) N (cid:0) − ∆ + R4 + V (cid:1) ≤ c ( n ) e − µ Z M V n − dv, for any function V ∈ L loc ( M ) , where V − := max { , − V } ∈ L n/ ( M ) is the non-positive part of V , and N ( A ) is the number of non-positive L -eigenvalues of theoperator A , counting multiplicity.Remark . For (III), (IV) and (VI), we need to assume the existence of Schr¨odinger heatkernel on complete shrinkers because our proof will be involved with the Schr¨odinger heatkernel. For compact shrinkers, the Schr¨odinger heat always exists; but for the non-compactcase, we only know that it exists when scalar curvature is bounded (see [55]).
Remark . We point out that (III) is equivalent to (1.5). Indeed, (III) ⇒ (1.5) is obvious,while (1.5) ⇒ (III) due to the work [55]. We also point out that (IV) is equivalent to (1.7).Indeed, from Theorem 1.1, we first have (IV) ⇒ (III), and from the work [55], we then know(III) ⇒ (1.7). Combining two parts finally yields (IV) ⇒ (1.7). The converse is trivial. EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 5
Remark . Estimates in (III) and (IV) are both sharp; see [55]. In addition (1.10) is alsosharp in some sense. Indeed on Gaussian shrinker ( R n , δ ij , | x | ), where δ ij is the standardflat Euclidean metric, we know R = 0 and µ = 0. For a large parameter α , replacing V by αV in (1.10), α − n N ( − ∆ + αV ) ≤ c ( n ) Z R n V n − dv. In turns out that for any potential V with V − ∈ L n/ ( R n ) and V ∈ L loc ( R n ), we have theWeyl asymptotics lim α →∞ α − n N ( − ∆ + αV ) = (2 √ π ) − n Γ(1 + n ) Z R n V n − dv. This indicates that (1.10) is sharp in order in α for a class function of V .The proof strategy of Theorem 1.1 is as follows. Li and Wang [39] proved that (II) holdson shrinkers and they confirmed that (II) ⇒ (I). The author [55] proved that (II) ⇒ (III) ⇒ (IV). The rest part is the following.(1) We will apply the Jensen inequality to confirm (I) ⇒ (II).(2) We shall apply the Davies’ argument [17] and the Markov semigroup to reprove Li-Wang’s Sobolev inequality (1.8), i.e., (III) ⇒ (I). In particular we will talk aboutthe scope of Sobolev constant, which will be useful to study gap theorems.(3) We will apply the Schr¨odinger heat kernel and the level set method to prove (IV) ⇒ (III) by the approximation argument.(4) We will use the H¨older inequality to prove (I) ⇒ (V).(5) By the approximation argument, we only need to apply the Schr¨odinger heat kerneland analytical technique to prove (V) ⇒ (III) with Dirichlet condition.(6) We will apply Schr¨odinger heat kernel properties and some functional theory to prove(I) ⇔ (VI).We remark that the proof of (III) ⇒ (I) will be separately provided in Section 2; the proofof (I) ⇔ (VI) will be given in Section 3; the rest cases will be discussed in Section 4.As applications, we will apply the Sobolev inequality of shrinkers to give integral gapresults for the Weyl tensor on compact shrinkers by adapting the proof strategy of [10].To state the result, we fix some notations. We denote by W , ◦ Ric and V ( M ) the Weyltensor, traceless Ricci tensor and the volume of manifold ( M, g ) respectively. The norm ofa ( k, s )-tensor T of ( M, g ) is defined by | T | g := g i m · · · g i k m k g j n ...g j s j s T j ...j s i ...i k T n ...n s m ...m k . Theorem 1.5.
Let ( M, g, f ) be an n -dimensional, ≤ n ≤ , compact shrinker. If Z M (cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric ◦ g (cid:12)(cid:12)(cid:12) n dv ! n + √ n − s n − n − ! V ( M ) n < √ n − n r n − n − ! e µn C ( n ) , where ◦ denotes the Kulkarni-Nomizu product and C ( n ) is the constant in the Sobolev in-equality (1.8) , then ( M, g, f ) is isometric to a quotient of the round sphere. JIA-YONG WU
Remark . Chang, Gursky and Yang [12] obtained integral gap results for compact man-ifolds in terms of the Yamabe constant. Catino [10] proved some integral gap results forcompact shrinkers, which was later improved in [21]. Our result involves the the Sobolevconstant of shrinkers rather than the Yamabe constant.The main ingredients of proving Theorem 1.5 are Bochner-Weitzenb¨ock type formulas forthe norm of curvature tensors, curvature algebraic inequalities and Kato type inequalities.The above pinching assumption is not true when n ≥
9. Indeed we will see that constant C ( n ) in Sobolev inequality (1.8) cannot be sufficiently small (see Remark 2.5), i.e., C ( n ) ≥ n − n ( n − πe . This range obviously affects the dimensional valid of the pinching assumption. On the otherhand, algebraic curvature inequalities and elliptic equation of the norm of traceless Riccitensor also restrict the choice of dimension n , such as inequality (5.5) in Section 5.In particular, inspired by Cao-Tran’s result [8], we apply the Sobolev inequality of shrinkerto get an integral gap result for the half Weyl tensor on compact four-dimensional shrinkers. Theorem 1.7.
Let ( M, g, f ) be a four-dimensional oriented compact shrinker. Let C (4) denote the constant in the Sobolev inequality (1.8) in ( M, g, f ) . If (cid:18)Z M | W ± | dv (cid:19) < e µ √ C (4) and Z M | δW ± | dv ≤ Z M R | W ± | dv, then W ± ≡ and hence ( M , g, f ) is isometric to a finite quotient of the round sphere orthe complex projective space.Remark . For relevant notations, see Section 6. Gursky [25] proved an integral pinchingresult for four-dimensional Einstein manifolds involving the norm of half Weyl tensor interms of the Euler characteristic and the signature. Cao and Tran [8] generalized Gursky’sresult to shrinkers. Our gap result depends on the Sobolev constant of shrinkers rather thantopological invariants.Inspired by Catino’s result [8], if we use the Yamabe constant instead of the Sobolevinequality, then we have another integral gap result.
Theorem 1.9.
Let ( M, g, f ) be a four-dimensional oriented compact shrinker satisfying (1.1) . If Z M | W ± | dv + 12 Z M | ◦ Ric | dv ≤ Z M R dv and Z M | δW ± | dv ≤ Z M R | W ± | dv, then ( M, g, f ) is isometric to a finite quotient of the round sphere or the complex projectivespace. EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 7
Remark . In Theorems 1.7 and 1.9, the first assumption is an pinching condition of halfWeyl tensor; while the second assumption is an pinching condition of harmonic half Weyltensor. It is an interesting question if the second assumption is unnecessary.There are many gap results for Einstein manifolds, Ricci solitons and closed manifolds;such as Catino and Mastrolia [11], Hebey and Vaugon [32], Li and Wang [38, 39], Munteanuand Wang [43], Petersen and Wylie [47], Singer [51], Tran [52], Zhang [63] and their refer-ences. In this paper we provide a different gap criterion, which depends on the constant C ( n ) of Sobolev inequality. It is an interesting question to estimate a best upper bound of C ( n ) on shrinkers.The structure of this paper is the following. In Section 2, we will recall some basicresults about algebraic inequalities of curvature tensors and some formulas of shrinkers.In particular, we will reprove the Sobolev inequality by the Schr¨odinger heat kernel upperbound. Meanwhile, we will discuss the best Sobolev constant of shrinkers. In Section 3,we will apply the Schr¨odinger heat kernel to study the equivalence between (I) and (VI) ofTheorem 1.1. In Section 4, we will prove the rest cases of Theorem 1.1. In Section 5, we willapply the Sobolev inequality of shrinkers and Weitzenb¨ock formulas for curvature tensorsto prove Theorem 1.5. In Section 6, we will study the gap results for half Weyl tensor. Weshall prove Theorems 1.7 and 1.9. Acknowledgement . This work is supported by the NSFC (11671141) and the NaturalScience Foundation of Shanghai (17ZR1412800).2.
Decomposition and Sobolev inequality
In this section we first give a brief introduction of curvature notations of the Riemannianmanifold ( M n , g ) and some algebraic inequalities of curvature tensors. Then we review somegeometric equations and formulas about shrinkers, especially for the Sobolev inequality andits explicit coefficient. These results will be used in the following sections. For more relatedresults, see [39], [55].We use g ij to be the local components of metric g and its inverse by g ij . Let Rm bethe (4 ,
0) Riemannian curvature tensor, whose local components denoted by R ijkl . Let Ricdenote the Ricci curvature with local components R ik = g jl R ijkl , and let R = g ik R ik be thescalar curvature. The traceless Ricci tensor is denoted by ◦ Ric = Ric − n R g, whose local components ◦ R ik = R ik − n R g ik . When n ≥
4, the Weyl tensor W is defined by the orthogonal decomposition W = Rm − R2 n ( n − g ◦ g − n − ◦ Ric ◦ g, where ◦ denotes the Kulkarni-Nomizu product for two symmetric tensors A and B , whichis defined as: ( A ◦ B ) ijkl = A ik B jl − A il B jk − A jk B il + A jl B ik . JIA-YONG WU
In local coordinates, we can write W as W ijkl =R ijkl − n − g ik R jl + g jl R ik − g il R jk − g jk R il )+ 1( n − n −
2) R( g ik g jl − g il g jk ) . The Weyl tensor has the same algebraic symmetries as the Riemannian curvature tensor. Itis well-known that the Weyl tensor is totally trace-free and it is conformall invariant: W ( e ϕ g ) = e ϕ W ( g )for any smooth function ϕ on M .In [10], Catino proved two algebraic curvature inequalities for any n -dimensional Rie-mannian manifold, which will be used in the gap theorems. Lemma 2.1.
Each n -dimensional Riemannian manifold ( M n , g ) satisfies the estimate (cid:12)(cid:12)(cid:12)(cid:12) − W ijkl ◦ R ik ◦ R jl + 2 n − ◦ R ij ◦ R jk ◦ R ik (cid:12)(cid:12)(cid:12)(cid:12) ≤ s n − n − | W | + 8 | ◦ Ric | n ( n − | ◦ Ric | . Lemma 2.2.
On an n -dimensional Riemannian manifold ( M n , g ) , there exists a positiveconstant c ( n ) such that W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij ≤ c ( n ) | W | . We can take c (4) = √ , c (5) = 1 , c (6) = √ √ and c ( n ) = for n ≥ . On shrinker (
M, g, f ), by Proposition 2.1 in [19], we have the following basic formulas,which will be also used in the proof of gap theorems.
Lemma 2.3.
Let ( M, g, f ) be an n -dimensional complete shrinker. Then, ∆ f = n − R , ∆ f R = R − | Ric | , ∆ f R ik = R ik − W ijkl R jl + 2( n − n − × (cid:16) R g ik − n RR ik + 2( n − g mn R im R nk − n − | Ric | g ik (cid:17) , where ∆ f := ∆ − ∇ f · ∇ . In the end of this section, we will give the following Sobolev inequality of shrinkers, whichwas proved by Li and Wang [39] by using the logarithmic Sobolev inequality. Here we shallreprove it by using the upper bound of Schr¨odinger heat kernel. Meanwhile we will providean explicit Sobolev constant and discuss its range. This constant will play a key role inproving gap results of shrinkers.
EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 9
Lemma 2.4.
Let ( M, g, f ) be an n -dimensional complete shrinker. Then for each compactlysupported locally Lipschitz function ϕ in M , (2.1) (cid:18)Z M ϕ nn − dv (cid:19) n − n ≤ C ( n ) e − µn Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv for some dimensional constant C ( n ) . In particular, we can take C ( n ) = 1 π (cid:18) n − (cid:19) n . Remark . For an n -sphere S n of radius p n −
1) with its standard metric, we haveRic = g . Recall that on S n , Aubin [1] (see also Proposition 4.21 in [31]) proved that forany ϕ ∈ W , ( S n ),(2.2) (cid:18)Z S n ϕ nn − dv (cid:19) n − n ≤ n − n ( n − V ( S n ) − /n Z S n |∇ ϕ | dv + V ( S n ) − /n Z S n ϕ dv. This inequality is optimal in the sense that the two constants n − n ( n − V ( S n ) − /n and V ( S n ) − /n can not be lowered. On the other hand, from (1.3), we see thatR = f = n πe ) − n V ( S n ) = e µ . Substituting them into (2.1) and comparing with (2.2), we easily conclude that C ( n ) ≥ n − n ( n − πe . Proof of Lemma 2.4.
We essentially follow the argument of Davies [17] (see also [39, 62]).Since H R = H R ( x, y, t ) is the Schr¨odinger heat kernel of the operator − ∆ + R4 , then Z M H R ( x, y, t ) dv ( y ) ≤ . In [55] we proved an upper of the Schr¨odinger heat kernel H R ( x, y, t ) ≤ e − µ (4 πt ) n for all x, y ∈ M and t >
0. In the following we will use this estimate to prove (2.1).Now using the H¨older inequality, for any u ( x ) ∈ L ( M ), we have k H R ∗ u k ∞ = sup x ∈ M (cid:12)(cid:12)(cid:12)(cid:12)Z M H R ( x, y, t ) u ( y ) dv ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x ∈ M (cid:18)Z M ( H R ) ( x, y, t ) dv ( y ) (cid:19) · k u k . By (1.5), we further have k H R ∗ u k ∞ ≤ e − µ (4 πt ) n sup x ∈ M (cid:18)Z M H R ( x, y, t ) dv ( y ) (cid:19) · k u k ≤ c / t n/ k u k , where c = e − µ (4 π ) n . Similarly, by the H¨older inequality, for any u ( x ) ∈ L q ( M ), q ∈ [1 , n ) and e q = q/ ( q − k H R ∗ u k ∞ ≤ sup x ∈ M (cid:18)Z M ( H R ) e q ( x, y, t ) dv ( y ) (cid:19) / e q · k u k q ≤ sup x ∈ M (cid:18) sup y ∈ M ( H R ) q − ( x, y, t ) Z M H R ( x, y, t ) dv ( y ) (cid:19) / e q · k u k q ≤ sup x ∈ M (cid:18) sup y ∈ M ( H R ) q − ( x, y, t ) (cid:19) / e q · sup x ∈ M (cid:18)Z M H R ( x, y, t ) dv ( y ) (cid:19) / e q · k u k q . Using (1.5) again, for any u ( x ) ∈ L q ( M ) and q ∈ [1 , n ), we finally get(2.3) k H R ∗ u k ∞ ≤ c /q t n q k u k q . Now we consider the integral operator L := (cid:18)q − ∆ + R4 (cid:19) − . Since − ∆ + R4 is a self-adjoint operator, by the eigenfunction expansion, for any u ( x ) ∈ C ∞ ( M ), we have ( Lu )( x ) = Γ(1 / − Z ∞ t − (cid:0) e (∆ − R / t u (cid:1) ( x, t ) dt = Γ(1 / − Z ∞ t − (cid:0) H R ∗ u (cid:1) ( x, t ) dt, where e (∆ − R / t u denotes the semigroup of H R ∗ u . Fix T >
0, which will be determinedlater and let( Lu )( x ) = Γ( ) − Z T t − (cid:0) H R ∗ u (cid:1) ( x, t ) dt + Γ( ) − Z ∞ T t − (cid:0) H R ∗ u (cid:1) ( x, t ) dt := ( L u )( x ) + ( L u )( x ) . For any λ >
0, we see that(2.4) (cid:12)(cid:12) { x (cid:12)(cid:12) | ( Lu )( x ) | ≥ λ } (cid:12)(cid:12) ≤ (cid:12)(cid:12) { x (cid:12)(cid:12) | ( L u )( x ) | ≥ λ/ } (cid:12)(cid:12) + (cid:12)(cid:12) { x (cid:12)(cid:12) | ( L u )( x ) | > λ/ } (cid:12)(cid:12) . EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 11
By (2.3) and the definition of L u , since Γ( ) = √ π , we have k L u k ∞ ≤ Γ(1 / − Z ∞ T t − c /q t n q k u k q ! dt = 2 qc /q ( n − q ) √ π · T − n q k u k q . We now choose T such that λ qc /q ( n − q ) √ π · T − n q k u k q . Then the set { x (cid:12)(cid:12) | ( L u )( x ) | > λ/ } = ∅ and (2.4) becomes (cid:12)(cid:12) { x (cid:12)(cid:12) | ( Lu )( x ) | ≥ λ } (cid:12)(cid:12) ≤ (cid:12)(cid:12) { x (cid:12)(cid:12) | ( L u )( x ) | ≥ λ/ } (cid:12)(cid:12) ≤ ( λ/ − q Z M | ( L u )( x ) | q dv ( x ) . We will estimate the right hand side of the above inequality. By the Minkowski inequalityfor two measured spaces and the H¨older inequality, we get that k L u k q ≤ Γ( ) − Z T t − k H R ∗ u ( · , t ) k q dt ≤ Γ( ) − Z T t − sup x ∈ M k H R ( x, · , t ) k · k u k q dt ≤ √ π T / k u k q . Hence, (cid:12)(cid:12) { x (cid:12)(cid:12) | ( Lu )( x ) | ≥ λ } (cid:12)(cid:12) ≤ (cid:18) √ π (cid:19) q λ − q T q/ k u k qq . According to the above choice of T , we have (cid:12)(cid:12) { x (cid:12)(cid:12) | ( Lu )( x ) | ≥ λ } (cid:12)(cid:12) ≤ c ( n, q ) c qn − q λ − r k u k rq , where r = qn/ ( n − q ) and c ( n, q ) = (cid:18) √ π (cid:19) nqn − q (cid:18) qn − q (cid:19) q n − q . We see that for all q ∈ [1 , n ), the linear operator L could map the space L q ( M ) into theweak L r ( M ) space. That is, k Lu k r,w ≤ c ( n, q ) r c n k u k q = 4 √ π (cid:18) qn − q (cid:19) qn c n k u k q . For any 0 < ǫ <<
1, letting q = q − ǫ , q = q + ǫ , r i = q i n/ ( n − q i ) ( i = 1 , k Lu k r i ,w ≤ √ π (cid:18) q i n − q i (cid:19) qin c n k u k q i . Applying the Marcinkiewicz interpolation theorem to the above case, for any 0 < t <
1, weget that k Lu k b ≤ √ π "(cid:18) q n − q (cid:19) q n t "(cid:18) q n − q (cid:19) q n − t c n k u k a , where 1 a = tq + 1 − tq , b = tr + 1 − tr . Since the coefficient is continuous with respect to ǫ at ǫ = 0, letting ǫ →
0+ and choosing q = 2 and p = 2 n/ ( n − a → b → n/ ( n −
2) and we finally get k Lu k p ≤ √ π (cid:18) n − (cid:19) n c n k u k , where c = e − µ (4 π ) n . Let ϕ = Lu and then u = L − ϕ and k u k = h L − ϕ, L − ϕ i = h L − ϕ, ϕ i = (cid:10) − ∆ ϕ + R4 ϕ, ϕ (cid:11) = Z M (cid:0) |∇ ϕ | + R4 ϕ (cid:1) dv. Substituting this into the above inequality proves the theorem. (cid:3)
In the above proof course , if we let q = 3, q = 1, t = 3 / r = nn − and r = nn − , wecan also take C ( n ) = 1 π (cid:18) n − (cid:19) n (cid:18) n − (cid:19) n . Obviously, when 4 ≤ n ≤
14, this constant is bigger than the one in Lemma 2.4 but when n ≥
15, it is reverse. Naturally it is to ask
Question . For a complete gradient shrinking Ricci soliton ( M, g, f ) , specially for the com-pact case, What is the best constant C ( n ) ? Rozenblum-Cwikel-Lieb inequality
It is well-known that the spectrum of the Laplacian − ∆ on Riemannian manifold ( M, g )is contained in the internal [0 , ∞ ]. This can be proved by taking the Fourier transformand using the Plancherel theorem. If one considers the Schr¨odinger operator − ∆ + V forsome function V on M , then − ∆ + V may have some negative spectrum. However, if wehave some restriction on V , like decay conditions at infinity, we may hope that the essentialspectra of − ∆ + V and − ∆ coincide. In this case, the negative spectrum of − ∆ + V isa discrete set with possibly an accumulation point at 0 (if 0 is indeed the bottom of thespectrum of − ∆). It is an important question in mathematical physics to estimate the EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 13 number of these negative eigenvalues. One of beautiful results about this question is theRozenblum-Cwikel-Lieb (RCL) inequality(3.1) N ( − ∆ + V ) ≤ c ( n ) Z M V n − dv, where V − is the negative part of function V ∈ L loc ( M ), and N ( A ) is the number of non-positive L -eigenvalues of the operator A . The RCL inequality was first established byRozenblum [49], and it was independently found by Lieb [40] and Cwikel [16] for n ≥ M, g, f ), one may would like to consider the special Schr¨odinger operator − ∆ R := − ∆ + R4 instead of the usual Laplacian. Since scalar curvature R ≥ M, g, f ), by the RCLinequality, it is easy to see that its eigenvalues are all nonnegative. If one considers aperturbation of − ∆ R by a real-valued potential V and defines another Schr¨odinger operator − ∆ R + V , then the nonnegative property of eigenvalues is not necessarily satisfied. Naturallyone may ask: What assumption on function V will imply a bound on the number of negative eigenvaluesof − ∆ R + V on shrinkers? In the following we will give an answer to this question. i.e., Theorem 1.1: (I) ⇒ (VI).To prove this result, we start with an important proposition, which is a key step of provingthe RCL type inequality on shrinkers. Proposition 3.1.
Let D be a bounded domain in a shrinker ( M n , g, f ) , where n ≥ .Assume that q ( x ) is a positive function defined on D . Let λ k be the k -th eigenvalue of theequation − ∆ R φ ( x ) = λq ( x ) φ ( x ) on D with the Dirichlet boundary condition φ | ∂D ≡ . Then, λ n/ k Z D q n/ ( x ) dv ( x ) ≥ c ( n ) e µ k for some dimensional constant c ( n ) .Proof of Proposition 3.1. Inspired by the Li-Yau work [36], we consider the “heat” kernel ofthe parabolic operator − ∆ R /q + ∂ t on shrinker ( M, g, f ). Let { φ i ( x ) } ∞ i =1 be a set of orthonormal eigenfunctions such that − ∆ R φ i = λ i qφ i , where λ i denote the eigenvalues of the corresponding eigenfunctions { φ i ( x ) } ∞ i =1 . Then thekernel e H ( x, y, t ) of − ∆ R /q + ∂ t must have the following expression e H ( x, y, t ) = ∞ X i =1 e − λ i t φ i ( x ) φ i ( y ) . By the property of Schr¨odinger heat kernel H R ( x, y, t ) (see [55]), we have e H ( x, y, t ) > D × D and e H ( x, y, t ) ≡ ∂D × ∂D for any t . At this time, the L -normis given by the weighted volume q ( x ) dv and Z D φ i ( x ) φ j ( x ) q ( x ) dv = δ ij Since(3.2) h ( t ) := ∞ X i =1 e − λ i t = Z D Z D e H ( x, y, t ) q ( x ) q ( y ) dv ( x ) dv ( y ) , then we have(3.3) dhdt = 2 Z D Z D e H ( x, y, t ) e H t ( x, y, t ) q ( x ) q ( y ) dv ( x ) dv ( y )= 2 Z D Z D e H ( x, y, t )∆ R y e H ( x, y, t ) q ( x ) dv ( y ) dv ( x )= − Z D Z D (cid:18) |∇ e H ( x, y, t ) | + R4 e H ( x, y, t ) (cid:19) q ( x ) dv ( y ) dv ( x ) , where we used ∆ R y q ( y ) − ∂ t ! e H ( x, y, t ) = 0 . On the other hand, using the Cauchy-Schwarz inequality, we have(3.4) h ( t ) = "Z D q ( x ) (cid:18)Z D e H nn − ( x, y, t ) dv ( y ) (cid:19) n − n dv ( x ) nn +2 × "Z D q ( x ) (cid:18)Z D e H ( x, y, t ) q n +24 ( y ) dv ( y ) (cid:19) dv ( x ) n +2 . Let us now analyze the above inequality. Consider the quantity Q ( x, t ) := Z D e H ( x, y, t ) q n +24 ( y ) dv ( y )and it satisfies (cid:18) ∆ R x q ( x ) − ∂ t (cid:19) Q ( x, t ) = 0with Q ( x, t ) ≡ ∂D for t > Q ( x,
0) = q n − ( x ) . We observe that ∂ t Z D q ( x ) Q ( x, t ) dv ( x ) = 2 Z D q ( x ) Q ( x, t ) ∂ t Q ( x, t ) dv ( x )= 2 Z D Q ( x, t )∆ R x Q ( x, t ) dv ( x )= − (cid:18)Z D |∇ x Q ( x, t ) | + R4 Q ( x, t ) (cid:19) dv ( x ) ≤ , EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 15 where we used the scalar curvature R ≥ Z D q ( x ) Q ( x, t ) dv ( x ) ≤ Z D q ( x ) Q ( x, dv ( x )= Z D q n/ ( x ) dv ( x ) . Using this, from (3.4) we have(3.5) h n +2 n ( t ) (cid:18)Z D q n/ ( x ) dv (cid:19) − n ≤ Z D q ( x ) (cid:18)Z D e H nn − ( x, y, t ) dv ( y ) (cid:19) n − n dv ( x ) . Recall that the Sobolev inequality (1.8) of shrinkers by letting ϕ = e H ( x, y, t ) says that (cid:18)Z D | e H | nn − dv ( y ) (cid:19) n − n ≤ C ( n ) e − µn Z D (cid:16) |∇ e H | + R e H (cid:17) dv ( y ) . Combining this with (3.5) and (3.3) yields dhdt ≤ − e µn C ( n ) (cid:18)Z D q n/ ( x ) dv ( x ) (cid:19) − n · h n +2 n ( t ) . Dividing this by h n +2 n ( t ) and integrating with respect to t , h ( t ) ≤ ( nC ( n )) n e − µ (cid:18)Z D q n/ ( x ) dv ( x ) (cid:19) t − n . Combining this with (3.2), we get ∞ X i =1 e − λ i t ≤ ( nC ( n )) n e − µ (cid:18)Z D q n/ ( x ) dv ( x ) (cid:19) t − n . Setting t = n λ k , we conclude that ∞ X i =1 e − nλi λk ≤ ( nC ( n )) n e − µ Z D q n/ ( x ) dv ( x ) (cid:18) n λ k (cid:19) − n . Noticing that ∞ X i =1 e − nλi λk ≥ ke − n/ , so we have λ n/ k Z D q n/ ( x ) dv ( x ) ≥ c ( n ) e µ k for some dimensional constant c ( n ). (cid:3) Now we will apply Proposition 3.1 to prove Theorem 1.1: (I) ⇒ (VI). Proof of Theorem 1.1: (I) ⇒ (VI). By the monotonicity of N (cid:0) − ∆ R + V (cid:1) with respect tothe function V ( x ) on shrinker ( M, f, g ), we may assume V ( x ) ≤ V ( x ) by − V − ( x ). Moreover − V − ( x ) can be approximated by a sequence of strictly negative function. So we can assume V ( x ) < x ∈ M . By the exhausting argument (see Lemma 3.2 in[45]), we only need to prove N (cid:0) − ∆ R + V (cid:1) ≤ c ( n ) Z D V n − dv for the equation(3.6) (cid:0) − ∆ R + V (cid:1) φ = λφ with φ | ∂D ≡ D ∈ M .It is easy to see that the number of non-positive eigenvalues N (cid:0) − ∆ R + V (cid:1) for (3.6),counting multiplicity, is equal to the number of eigenvalues less than 1 for the case inProposition 3.1 by considering q ( x ) = − V ( x ) . Indeed, since V ( x ) <
0, from the relation R D (cid:0) |∇ φ | + R4 φ (cid:1) dv + R D V φ dv R D φ dv = R D | V | φ dv R D φ dv R D (cid:0) |∇ φ | + R4 φ (cid:1) dv R D | V | φ dv − ! , we conclude that the dimension of the subspace on which the left hand side is non-positiveis equal to the dimesion of the subspace on which the quadratic form R D (cid:0) |∇ φ | + R4 φ (cid:1) dv R D | V | φ dv associated to Proposition 3.1 is not more than 1. Now we let λ k be the greatest eigenvaluewhich is not more than 1. By Proposition 3.1, we have Z D | V | n/ dv ( x ) ≥ λ n/ k Z D | V | n/ dv ( x ) ≥ c ( n ) e µ k ≥ c ( n ) e µ · N ( − ∆ R + V ) , which completes the proof of (I) ⇒ (VI). (cid:3) Next we will prove the easy implication (VI) ⇒ (I). Proof of Theorem 1.1: (VI) ⇒ (I). We assume (1.10) holds for all potentials V ∈ C ∞ ( M ).Then for all V ∈ C ∞ ( M ) satisfying k V k n/ < c ( n ) e − µn , we know that − ∆ R + V is a non-negative operator. That is, Z M |∇ ϕ | + R4 ϕ dv + Z M V ϕ dv ≥ V and all ϕ ∈ C ∞ ( M ). In other words, Z M |∇ ϕ | + R4 ϕ dv ≥ sup V ∈ C ∞ ( M ) (cid:26)Z M − V ϕ dv (cid:27) satisfying k V k n/ < c ( n ) e − µn . EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 17
Since the dual of L n/ ( M ) is L n/ ( n − ( M ), the above functional inequality implies c ( n ) e − µn Z M (cid:18) |∇ ϕ | + R4 ϕ (cid:19) dv ≥ (cid:18)Z M ϕ nn − dv (cid:19) n − n and Theorem 1.1 (I) follows. (cid:3) Equivalence of geometric inequalities
In this section, we will give rest proofs of Theorem 1.1. This part of theorem mainly saysthat the (logarithmic) Sobolev inequality, the Schr¨odinger heat kernel upper bound, theFaber-Krahn inequality and the Nash inequality are all equivalent up to possible differentconstants. Notice that (II) ⇒ (I) was proved in [39]; (II) ⇒ (III) ⇒ (IV) was proved in[55]. So we only need to consider the following cases: (I) ⇒ (II), (III) ⇒ (I), (IV) ⇒ (III),(I) ⇒ (V), (V) ⇒ (III). Proof of Theorem 1.1. (I) ⇒ (II): We may assume (1.8) holds on shrinkers. That is, foreach compactly supported locally Lipschitz function ϕ in ( M, g, f ), (cid:18)Z M ϕ nn − dv (cid:19) n − n ≤ C ( n ) e − µn Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv for some dimensional constant C ( n ). Given function ϕ with k ϕ k = 1, we introduce theweighted measure dµ = ϕ dv on shrinker ( M, g, f ), then R M dµ = 1. Since function ln G isconcave with respect to parameter G , letting G = ϕ q − , where q = nn − , and applying theJensen inequality Z M ln Gdµ ≤ ln (cid:18)Z M Gdµ (cid:19) , we have that Z M (ln ϕ q − ) ϕ dv ≤ ln (cid:18)Z M ϕ q − ϕ dv (cid:19) = ln k ϕ k qq , That is, Z M ϕ ln ϕdv ≤ qq − k ϕ k q = n k ϕ k q . Combining this with the Sobolev inequality (1.8), we get Z M ϕ ln ϕ dv ≤ n k ϕ k q ≤ n (cid:20) C ( n ) e − µn Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv (cid:21) = n C ( n ) − µ + n (cid:20)Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv (cid:21) . Using an elementary inequality: ln x ≤ σx − (1 + ln σ )for any σ >
0, the above estimate can be further reduced to Z M ϕ ln ϕ dv ( x ) ≤ n C ( n ) − µ + nσ Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv − n σ ) . Setting τ = nσ , we obtain Z M ϕ ln ϕ dv ≤ τ Z M (cid:0) |∇ ϕ | + R ϕ (cid:1) dv − µ − n − n πτ ) + n neπ · C ( n ))and hence (b) follows with possible different constants.(III) ⇒ (I): Since (III) implies (1.5) with different constants, then (1.5) further implies(I) by following the proof of Lemma 2.4 in Section 2.(IV) ⇒ (III): Since (1.5) is equivalent to (III), we only need to apply (IV) to prove (1.5).By the approximation argument, we only need to prove (1.5) for the Dirichlet Schr¨odingerheat kernel H RΩ ( x, y, t ) of any relatively compact set Ω in ( M, g, f ). In fact, let Ω i , i = 1 , , ... ,be a sequence of compact exhaustion of M such that Ω i ⊂ Ω i +1 and ∪ i Ω i = M . If we areable to prove (1.5) for the Dirichlet Schr¨odinger heat kernel H RΩ i ( x, y, t ) for any i , then theresult follows by letting i → ∞ .For a fixed point y ∈ Ω, let u = u ( x, t ) = H RΩ ( x, y, t ) and consider the integral I ( t ) := Z Ω u ( x, t ) dv. Then,(4.1) I ′ ( t ) = 2 Z Ω uu t dv = − Z Ω (cid:18) |∇ u | + 14 R u (cid:19) dv. For any positive number s , we have u ≤ ( u − s ) + 2 su and therefore, I ( t ) ≤ Z Ω ( u − s ) dv + 2 s Z Ω udv. Now for fixed s, t >
0, consider the set D ( s, t ) := { x | x ∈ M, u ( x, t ) > s } and its the first eigenvalue λ ( D ( s, t )) = inf = ϕ ∈ C ∞ ( D ( s,t )) R D ( s,t ) ( |∇ ϕ | + R4 ϕ ) dv R D ( s,t ) ϕ dv . EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 19
Letting ϕ = ( u − s ) + , then λ ( D ( s, t )) ( I ( t ) − s ) ≤ Z D ( s,t ) (cid:18) |∇ ( u − s ) + | + R4 (( u − s ) + ) (cid:19) dv ≤ Z Ω (cid:18) |∇ u | + R4 u (cid:19) dv. Note that in the above first inequality, we threw away a positive term and used theSchr¨odinger heat kernel property R Ω u ( x, t ) dv ≤
1. This property also indicates that V ( D ( s, t )) ≤ s − . On the other hand, by the Faber-Krahn inequality, we have λ ( D ( s, t )) ≥ nπe (cid:18) e µ V ( D ( s, t )) (cid:19) n ≥ nπe ( e µ · s ) n . We remark that if the set D ( s, t ) is not relatively compact, we can choose a sequence ofrelatively compact sets which converges to it such that the Faber-Krahn inequality remainsvalid for D ( s, t ). Combining the above two inequalities, we obtain I ( t ) ≤ e − µ/n nπ Z Ω (cid:18) |∇ u | + R4 u (cid:19) dv · s − /n + 2 s. Minimizing the right hand side of the above inequality, I ( t ) ≤ c ( n ) e − µn +2 (cid:20)Z Ω (cid:18) |∇ u | + R4 u (cid:19) dv (cid:21) nn +2 . Combining this with (4.1), we have I ′ ( t ) ≤ c ( n ) e µn I n +2 n . Integrating this from t/ t , I ( t ) ≤ c ( n ) e − µ t n/ for t >
0. In other words, we in fact get Z Ω H RΩ ( x, y, t ) H RΩ ( x, y, t ) dv ( y ) ≤ c ( n ) e − µ t n/ . By the Schr¨odinger heat kernel property, we indeed show that H RΩ ( x, x, t ) = Z Ω H RΩ ( x, y, t ) H RΩ ( y, x, t ) dv ( y ) ≤ c ( n ) e − µ t n/ . This further implies H RΩ ( x, y, t ) = Z Ω H RΩ ( x, z, t/ H RΩ ( z, y, t/ dv ( z ) ≤ (cid:18)Z Ω ( H RΩ ) ( x, z, t/ dv ( z ) (cid:19) / (cid:18)Z Ω ( H RΩ ) ( z, y, t/ dv ( z ) (cid:19) / = ( H RΩ ) / ( x, x, t )( H RΩ ) / ( y, y, t ) ≤ c ( n ) e − µ t n/ . Next we apply the same argument of proving Theorem 1.1 in [55] to get an upper boundwith a Gaussian exponential factor and finally (III) follows.(I) ⇒ (V): We remark that the Nash inequality can be viewed as an interpolation betweenthe H¨older inequality and the Sobolev inequality. Assume that ( M, g, f ) admits (1.8). Bythe H¨older inequality, for p = n +2 n − and p = n +24 , we have Z M ϕ dv = Z M ϕ nn +2 ϕ n +2 dv ≤ (cid:18)Z M ϕ nn +2 p dv (cid:19) /p (cid:18)Z M ϕ n +2 p dv (cid:19) /p = (cid:18)Z M ϕ nn − dv (cid:19) n − n +2 (cid:18)Z M | ϕ | dv (cid:19) n +2 and hence, k ϕ k n ≤ (cid:18)Z M ϕ nn − dv (cid:19) n − n (cid:18)Z M | ϕ | dv (cid:19) n . Combining this with the Sobolev inequality (1.8) gives the Nash inequality (1.9).(V) ⇒ (III): Using the approximation argument, it suffices to prove (1.6) for the DirichletSchr¨odinger heat kernel H RΩ ( x, y, t ) of any relatively compact set Ω in ( M, g, f ).For any y ∈ M , let ϕ = ϕ ( x, t ) = H RΩ ( x, y, t ). Then ∂∂t (cid:18)Z Ω ϕ dv (cid:19) = Z Ω ϕϕ t dv = Z Ω ϕ (∆ ϕ −
14 R ϕ ) dv = − Z Ω (cid:0) |∇ ϕ | + R ϕ (cid:1) dv. Scaling function ϕ such that k ϕ k = 1, by our assumption, we may assume the Nashinequality k ϕ k n ≤ c ( n ) e − µn Z Ω (cid:0) |∇ ϕ | + R ϕ (cid:1) dv. EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 21
Combining the above estimates, we have ∂∂t (cid:18)Z Ω ϕ dv (cid:19) ≤ − e µn c ( n ) k ϕ k n . Let F ( s ) := Z Ω ϕ ( x, s ) dv, where s ∈ (0 , t ]. Then ∂∂s F ( s ) ≤ − e µn c ( n ) F ( s ) n . Integrating it from t/ t yields − n (cid:16) F ( t ) − n − F ( t/ − n (cid:17) ≤ − e µn c ( n ) · t , which implies that F ( t ) ≤ [2 nc ( n )] n/ e − µ t n/ . This estimate is the same as I ( t ) in the proof of the case (IV) ⇒ (III). Therefore we onlyuse the same strategy of proving Theorem 1.1 in [55] to get an upper bound with a Gaussianexponential factor and finally prove (III). (cid:3) Gap result for Weyl tensor
In this section we will prove Theorem 1.5 by using the arguments of [10, 21]. We firstrecall the elliptic equation of the norm of traceless Ricci tensor on shrinkers, which can bedirectly computed by Lemma 2.3 (see also Lemma 3.2 in [10]).
Lemma 5.1. If ( M n , g, f ) be an n -dimensional shrinker satisfying (1.2) , then
12 ∆ f | ◦ Ric | = |∇ ◦ Ric | + | ◦ Ric | − W ijkl ◦ R ik ◦ R jl + 4 n − ◦ R ij ◦ R jk ◦ R ik − n − n ( n −
1) R | ◦ Ric | . In the following we will apply the similar arguments of [10, 21] to prove gap theoremson shrinkers. In our case, we need to carefully deal with the constant C ( n ) of Sobolevinequality (1.8). Proof of Theorem 1.5.
By Lemma 5.1, using12 ∆ | ◦ Ric | = |∇| ◦ Ric || + | ◦ Ric | ∆ | ◦ Ric | and the Kato inequality |∇ ◦ Ric | ≥ |∇| ◦ Ric || at each point where | ◦ Ric | 6 = 0, we obtain(5.1) | ◦ Ric | ∆ | ◦ Ric | ≥ | ◦ Ric | − W ijkl ◦ R ik ◦ R jl + 4 n − ◦ R ij ◦ R jk ◦ R ki − n − n ( n −
1) R | ◦ Ric | + 12 h∇ f, ∇| ◦ Ric | i . To simplify the notation, we let u := | ◦ Ric | . Then for any positive number s , which will bedetermined later, by (5.1), we compute that u s ∆ u s = s ( s − u s − |∇ u | + su s − ∆ u = (cid:18) − s (cid:19) |∇ u s | + su s − u ∆ u ≥ (cid:18) − s (cid:19) |∇ u s | + su s + s (cid:18) − W ijkl ◦ R ik ◦ R jl + 4 n − ◦ R ij ◦ R jk ◦ R ki (cid:19) u s − − n − n ( n − s R u s + s u s − h∇ f, ∇ u i . Using Lemma 2.1, we further have u s ∆ u s ≥ (cid:18) − s (cid:19) |∇ u s | + su s − r n − n − s (cid:18) | W | + 8 u n ( n − (cid:19) u s − n − n ( n − s R u s + 12 h∇ f, ∇ u s i . Since M n is closed, integrating by parts over M n and using the equality∆ f = n − Rfrom Lemma 2.3, we have that0 ≥ (cid:18) − s (cid:19) Z M |∇ u s | dv + s Z M u s dv − r n − n − s Z M (cid:18) | W | + 8 u n ( n − (cid:19) u s dv − n − n ( n − s Z M R u s dv − Z M u s ∆ f dv = (cid:18) − s (cid:19) Z M |∇ u s | dv − r n − n − s Z M (cid:18) | W | + 8 u n ( n − (cid:19) u s dv − (cid:16) n − s (cid:17) Z M u s dv + n ( n − − n − s n ( n − Z M R u s dv. For 2 − /s >
0, by the Sobolev inequality of shrinker using ϕ = u s (5.2) Z M |∇ u s | dv ≥ e µn C ( n ) (cid:18)Z M u nsn − dv (cid:19) n − n − Z M R u s dv, the above inequality becomes0 ≥ (cid:18) − s (cid:19) e µn C ( n ) (cid:18)Z M u nsn − dv (cid:19) n − n − r n − n − s Z M (cid:18) | W | + 8 u n ( n − (cid:19) u s dv − (cid:16) n − s (cid:17) Z M u s dv + n ( n − − n − s n ( n − s Z M R u s dv. EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 23
By the H¨older inequality, for n − s ≥
0, we get that0 ≥ (cid:18) − s (cid:19) e µn C ( n ) − r n − n − s "Z M (cid:18) | W | + 8 u n ( n − (cid:19) n dv n − (cid:16) n − s (cid:17) V ( M ) n × (cid:18)Z M u nsn − dv (cid:19) n − n + n ( n − − n − s n ( n − s Z M R u s dv. Now we choose s = s n ( n − n − ∈ (cid:18) , n (cid:21) , and the last term of the above inequality vanishes. Moreover, notice that the curvatureintegral assumption of theorem is equivalent to(5.3) (cid:18) − s (cid:19) e µn C ( n ) − r n − n − s "Z M (cid:18) | W | + 8 u n ( n − (cid:19) n dv n − (cid:16) n − s (cid:17) V ( M ) n > , where we used the equality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric ◦ g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | W | + 8 n ( n − | ◦ Ric | due to the totally trace-free tensor W . Therefore, we conclude that | ◦ Ric | ≡
M, g, f )is Einstein.Now we have Ric = g . By (1.3), we know f = n πe ) − n V ( M ) = e µ and the pinching condition (5.3) reduces to(5.4) (cid:18) − Z M | W | n dv (cid:19) n ≤ ǫ ( n ) := s n − n − (cid:20)(cid:18) s − s (cid:19) C ( n ) − πe + 4 − ns (cid:21) , where − R M denotes the average of the integration, i.e., − Z M | W | n dv = 1 V ( M ) Z M | W | n dv. By Remark 2.5, we see C ( n ) ≥ n − n ( n − πe and hence ǫ ( n ) ≤ s n − n − (cid:20)(cid:18) s − s (cid:19) n ( n − n −
1) + 4 − ns (cid:21) . Notice that the right hand side of the above inequality is nonnegative if(5.5) s ≥ n + p n + 8 n ( n − n − n − . Since s = q n ( n − n − , it is easy to check that the above inequality only holds only when4 ≤ n ≤
8. We then carefully calculate the constants as follows: ǫ (4) ≤ √ − ≈ . , ǫ (5) ≤ √ − √ ≈ . , ǫ (6) ≤ √ − √ ≈ . ,ǫ (7) ≤ √ − √ √ ≈ . , ǫ (8) ≤ √ − √ ≈ × − . Obviously, these constants in (5.4) are strictly smaller than those in the following Proposition5.2, and hence (
M, g, f ) is isometric to a quotient of the sphere. (cid:3)
The following result is an gap result for Einstein manifolds, which was essentially provedby Catino [10]. The present version of pinching constants is a little better than Theorem3.3 in [10] and seems to be more suitable to our applicable purpose.
Proposition 5.2.
Let ( M n , g ) be an n -dimensional Einstein manifold with Ric = kg , where k > is a constant. There exists a positive constant ǫ ( n ) depending only on n such that if (cid:18) − Z M | W | n dv (cid:19) n < ǫ ( n ) , where − R M | W | n dv = V ( M ) R M | W | n dv , then ( M n , g ) is isometric to a quotient of the roundsphere with radius q n − k . We can take ǫ (4) = √ k , ǫ (5) = k , ǫ (6) = √ √ k , ǫ (7) = k , ǫ (8) = k , ǫ (9) = k and ǫ ( n ) = n n − k if n ≥ .Proof of Proposition 5.2. Following the argument in [32, 10], we have the Bochner typeformula for | W | ,12 ∆ | W | = |∇ W | + 2 k | W | − (cid:18) W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij (cid:19) . Since 12 ∆ | W | = |∇| W || + | W | ∆ | W | , then we have | W | ∆ | W | = |∇ W | − |∇| W || + 2 k | W | − (cid:18) W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij (cid:19) . Using Lemma 2.2 and the refined Kato inequality |∇ W | ≥ n + 1 n − |∇| W || at every point where | W | 6 = 0, we get(5.6) | W | ∆ | W | ≥ n − |∇ W | + 2 k | W | − c ( n ) | W | , where c ( n ) is a dimensional constant, which is defined by c (4) = √ , c (5) = 1, c (6) = √ √ and c ( n ) = for n ≥ EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 25
Similar to the preceding computation, we consider the quantity u s := | W | s , where s is apositive number, which will be chosen later. Using (5.6), we compute u s ∆ u s = (cid:18) − s (cid:19) |∇ u s | + su s − u ∆ u ≥ (cid:18) − s (cid:19) |∇ u s | + 2 sn − u s − |∇ u | + 2 ksu s − c ( n ) su s +1 = (cid:18) − n − n − s (cid:19) |∇ u s | + 2 ksu s − c ( n ) su s +1 . Since M n is closed, integrating the above inequality over M n yields(5.7) 0 ≥ (cid:18) − n − n − s (cid:19) Z M |∇ u s | dv + 2 ks Z M u s dv − c ( n ) s Z M u s +1 dv. Recall that Ilias [34] proved the Sobolev inequality of manifolds satisfying Ric ≥ kg , where k >
0, (by letting f = | W | s = u s in [34]), which can be stated that in our setting (cid:18)Z M u nsn − dv (cid:19) n − n ≤ n − n ( n − k V ( M ) − /n Z M |∇ u s | dv + V ( M ) − /n Z M u s dv. Applying the H¨older inequality and the above Sobolev inequality, (5.7) becomes0 ≥ (cid:18) − n − n − s (cid:19) Z M |∇ u s | dv + 2 ks Z M u s dv − c ( n ) s (cid:18)Z M u n dv (cid:19) n (cid:18)Z M u nsn − dv (cid:19) n − n ≥ (cid:18) − n − n − s (cid:19) Z M |∇ u s | dv + 2 ks Z M u s dv − c ( n ) sV ( M ) − /n (cid:18)Z M u n dv (cid:19) n (cid:18) n − n ( n − k Z M |∇ u s | dv + Z M u s dv (cid:19) . By the proposition assumption, we have the following equivalent form (cid:18)Z M u n dv (cid:19) n < ǫ ( n ) V ( M ) /n . Therefore, for s >
0, if ǫ ( n ) satisfies ( − n − n − s − c ( n ) sǫ ( n ) ( n − n ( n − k ≥ , ks − c ( n ) ǫ ( n ) ≥ , we immediately have W ≡ g has constant positive sectional curvature. Here we giveexplicit constants such that the above two inequalities holds.When n = 4 and s = , since c (4) = √ , we can take ǫ (4) = √ k .When n = 5 and s = , since c (5) = 1, we can take ǫ (5) = k .When n = 6 and s = , since c (6) = √ √ , we can take ǫ (6) = √ √ k .When n = 7 and s = , since c (7) = , we can take ǫ (7) = k .When n = 8 and s = , since c (8) = , we can take ǫ (8) = k .When n = 9 and s = , since c (9) = , we can take ǫ (9) = k .When n ≥
10 and s = nn − , since c ( n ) = , we can take ǫ ( n ) = n n − k . (cid:3) Gap result for half Weyl tensor
In this section we will apply the similar argument of Section 5 to talk about an gapphenomenon for shrinkers under the integral condition of half Weyl tensor.Recall that, on an oriented 4-dimensional shrinker (
M, g, f ), the bundle of 2-forms ∧ M can be decomposed as a direct sum ∧ M = ∧ + M + ∧ − M, where ∧ ± M is the ( ± ⋆ : ∧ M → ∧ M. Let { e i } i =1 be an oriented orthonormal basis of tangent bundle T M . For any pair ( ij ),1 ≤ i = j ≤
4, let ( i ′ j ′ ) denote the dual of ( ij ), i.e., the pair such that e i ∧ e j ± e i ′ ∧ e j ′ ∈ ∧ ± M. In other words, ( iji ′ j ′ ) = σ (1234) for some even permutation σ ∈ S . For the Weyl tensor W , its (anti-)self-dual part is W ± ijkl = 14 ( W ijkl ± W ijk ′ l ′ ± W i ′ j ′ kl + W i ′ j ′ k ′ l ′ ) . It is easy to check that W ± ijkl = ± W ± ijk ′ l ′ = ± W ± i ′ j ′ kl = W ± i ′ j ′ k ′ l ′ = 12 ( W ijkl ± W ijk ′ l ′ ) . On shrinkers, we have the following Weitzenb¨ock formula for W ± (see [8] or its general-ization [60]), and it is useful for analyzing the structure of shrinkers. Lemma 6.1.
Let ( M, g, f ) be a four-dimensional shrinker satisfying (1.1) . Then
12 ∆ f | W ± | = |∇ W ± | + 2 λ | W ± | −
18 det W ± − h ( ◦ Ric ◦ ◦ Ric) ± , W ± i . Using Lemma 6.1, we can prove Theorem 1.7 in the introduction.
Proof of Theorem 1.7.
By Lemma 6.1, using the following algebraic inequality observed byCao and Tran [8] det W ± ≤ √ | W ± | and the Kato inequality |∇ W ± | ≥ |∇| W ± || at every point where | W ± | 6 = 0, we get12 ∆ f | W ± | ≥ |∇| W ± || + 2 λ | W ± | − √ | W ± | − h ( ◦ Ric ◦ ◦ Ric) ± , W ± i . Since 12 ∆ | W ± | = |∇| W ± || + | W ± | ∆ | W ± | , then we have | W ± | ∆ | W ± | ≥ λ | W ± | − √ | W ± | − h ( ◦ Ric ◦ ◦ Ric) ± , W ± i + 12 h∇ f, ∇| W ± | i . EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 27
Since M n is closed, integrating the above inequality and integrating by parts over M n , wehave(6.1) 0 ≥ Z M |∇| W ± || dv + 2 λ Z M | W ± | dv − √ Z M | W ± | dv − Z M h ( ◦ Ric ◦ ◦ Ric) ± , W ± i dv − Z M | W ± | ∆ f dv. Note that Cao and Tran (Corollary 5.8 in [8]) observed Z M h ( ◦ Ric ◦ ◦ Ric) ± , W ± i dv = 4 Z M | δW ± | dv, and hence the second assumption of theorem in fact is Z M h ( ◦ Ric ◦ ◦ Ric) ± , W ± i dv ≤ Z M R | W ± | dv. Using this, (6.1) becomes0 ≥ Z M |∇| W ± || dv + 2 λ Z M | W ± | dv − √ Z M | W ± | dv − Z M R | W ± | dv − Z M | W ± | ∆ f dv. Using the equality of shrinkers ∆ f = 4 λ − R , we further have(6.2) 0 ≥ Z M |∇| W ± || dv − √ Z M | W ± | dv + 14 Z M R | W ± | dv. By the Sobolev inequality of shrinker (1.8) by letting ϕ = | W ± | , Z M |∇| W ± || dv ≥ e µ C (4) (cid:18)Z M | W ± | dv (cid:19) − Z M R | W ± | dv, then (6.2) can be simplified as0 ≥ e µ C (4) (cid:18)Z M | W ± | dv (cid:19) − √ Z M | W ± | dv. Using the H¨older inequality,0 ≥ " e µ C (4) − √ (cid:18)Z M | W ± | dv (cid:19) M | W ± | dv (cid:19) . By the first assumption of theorem, we immediately get W ± ≡
0. Finally we apply theclassification of Chen and Wang [15] to conclude that the shrinker is isometric to a finitequotient of the round sphere or the complex projective space. (cid:3)
In the end of this section, following the argument of Catino [10], we can apply the Yamabeconstant to give another gap result, i.e., Theorem 1.9 in introduction.
Proof of Theorem 1.9.
Similar to the argument of Theorem 1.7, using the second assumptionof theorem, (6.1) can also be written as0 ≥ Z M |∇| W ± || dv + 2 λ Z M | W ± | dv − √ Z M | W ± | dv − Z M R | W ± | dv − Z M | W ± | ∆ f dv. Using the shrinker’s equality ∆ f = 4 λ − R , we obtain(6.3) 0 ≥ Z M |∇| W ± || dv − √ Z M | W ± | dv + 16 Z M R | W ± | dv. We will apply the Yamabe constant to estimate the first gradient term in the above inequal-ity. Recall that the Yamabe constant Y ( M, [ g ]) is defined by Y ( M, [ g ]) := inf ϕ ∈ W , ( M ) 4( n − n − R M |∇ ϕ | dv g + R M R ϕ dv g ( R M | ϕ | n/ ( n − dv g ) ( n − /n , where [ g ] denotes the conformal class of g . As we all known, Y ( M, [ g ]) is positive on acompact manifold if and only if there exits a conformal metric in [ g ] whose scalar curvatureis positive everywhere. Hence the compact shrinker has positive Yamabe constant Y ( M, [ g ]).If we let ϕ = | W ± | on a four-dimensional compact shrinker, then the Yamabe constant Y ( M, [ g ]) implies the following inequality Z M |∇| W ± || dv ≥ Y ( M, [ g ])6 (cid:18)Z M | W ± | dv (cid:19) − Z M R | W ± | dv. Using this, (6.3) can be reduced to0 ≥ Y ( M, [ g ])6 (cid:18)Z M | W ± | dv (cid:19) − √ Z M | W ± | dv. By the H¨older inequality, we have(6.4) 0 ≥ " Y ( M, [ g ])6 − √ (cid:18)Z M | W ± | dv (cid:19) M | W ± | dv (cid:19) . Recall that Gursky [24] proved the following estimate on a compact four-dimensional man-ifold Z M R dv − Z M | ◦ Ric | dv ≤ Y ( M, [ g ]) . Here this inequality is strict unless the manifold is conformally Einstein. Combining thiswith the first assumption of theorem, we have6 √ (cid:18)Z M | W ± | dv (cid:19) ≤ Y ( M, [ g ]) . Combining this with (6.4) we conclude that W ± ≡ M, g ) is conformally Einstein.When W ± ≡
0, by Theorem 1.7, ( M , g, f ) is isometric to a finite quotient of the roundsphere or the complex projective space. EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 29
When (
M, g ) is conformally Einstein, it is naturally Bach flat (see Proposition 4.78 in [2])and hence is Einstein (see Theorem 1.1 in [6]). Now since ( M , g ) is Einstein, combining thefirst pinching condition of theorem and a gap result of Gursky and Lebrun (see Theorem 1in [26]), we also get W ± ≡ M , g, f ) is also isometric to a finite quotient ofthe round sphere or the complex projective space. (cid:3) References
1. T. Aubin, Equations diff´erentielles non lin´eaires et probl`eme de Yamabe concernant la courbure scalaire,J. Math. Pures Appl. 55, (1976) 269-296.2. A. Besse, Einstein Manifolds, Springer-Verlag, Berlin-Heidelberg, 1987.3. B. Chow, P. Lu, B. Yang, Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons,C. R. Math. Acad. Sci. Paris 349 (23-24) (2011) 1265-1267.4. H.-D. Cao, Geometry of complete gradient shrinking Ricci solitons, Geometry and analysis. No. 1,227-246, Adv. Lect. Math. (ALM), 17, Int. Press, Somerville, MA, 2011.5. H.-D. Cao, B.-L. Chen, X.-P. Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differentialgeometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp.47-112.6. H.-D. Cao, Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), 1149-1169.7. H.-D. Cao, D. Zhou, On complete gradient shrinking Ricci solitons, J. Diff. Geom. 85 (2010), 175-186.8. X.-D. Cao, H. Tran, The Weyl tensor of gradient Ricci solitons, Geom. Topol. 20 (2016), 389-436.9. J. Carrillo, L. Ni, Sharp logarithmic Sobolev inequalities on gradient solitons and applications, Comm.Anal. Geom. 17 (2009), 721-753.10. G. Catino, Integral pinched shrinking Ricci solitons, Adv. Math. 303 (2016), 279-294.11. G. Catino, P. Mastrolia, Bochner-type formulas for the Weyl tensor on four-dimensional Einstein man-ifolds, Int. Math. Res. Not. IMRN 12 (2020), 3794-3823.12. S.-Y.A. Chang, M.J. Gursky, P.C. Yang, A conformally invariant sphere theorem in four dimensions,Publ. Math. Inst. Hautes ´Etudes Sci. 98 (2003), 105-143.13. J. Cheeger, G. Tian, Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math.Soc. 19 (2005), 487-525.14. B.-L. Chen, Strong uniqueness of the Ricci flow, J. Diff. Geom. 82 (2009), 363-382.15. X.-X. Chen, Y.-Q. Wang, On four-dimensional anti-self-dual gradient Ricci solitons, J. Geom. Anal.25(2), (2015), 1335-1343.16. M. Cwikel, Weak type estimates for singular values and the number of bound states of Schr¨odingeroperators, Ann. Math. 2 (1977), 93-100.17. E.B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, 92, Cambridge Uni-versity Press, Cambridge, 1989.18. A. Derdzi´nski, A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc. 134, (2006),3645-3648.19. M. Eminenti, G. La Nave, C. Mantegazza, Ricci solitons: the equation point of view, Manuscr. Math.127(3), (2008), 345-367.20. J. Enders, R. M¨uller, P. Topping, On Type-I singularities in Ricci flow, Comm. Anal. Geom. 19(2011),905-922.21. H.-P. Fu, L.-Q. Xiao, Rigidity theorem for integral pinched shrinking Ricci solitons, Monatsh. Math.183 (2017), 487-494.22. H.-B. Ge, W.-S. Jiang, ǫ -regularity for shrinking Ricci solitons and Ricci flows, Geom. Funct. Anal. 27(2017), 1231-1256.23. A. Grigor’yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, Vol.47, 2009.24. M.J. Gursky, Locally conformally flat four- and six-manifolds of positive scalar curvature and positiveEuler characteristic, Indiana Univ. Math. J. 43 (1994): 747-774.
25. M.J. Gursky, Four-manifolds with δW + = 0 and Einstein constants of the sphere, Math. Ann. 318(2000), 417-431.26. M.J. Gursky, C. Lebrun, On Einstein manifolds of positive sectional curvature, Ann. Global Anal. Geom.17 (4) (1999), 315-328.27. R. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics 71 (1988), 237-261.28. R. Hamilton, The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, Inter-national Press, Boston, vol. 2, (1995), 7-136.29. R. Haslhofer, R. M¨uller, A compactness theorem for complete Ricci shrinkers, Geom. Funct. Anal. 21(2011), 1091-1116.30. R. Haslhofer, R. M¨uller, A note on the compactness theorem for 4d Ricci shrinkers, Proc. Amer. Math.Soc. 143 (2015), 4433-4437.31. E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996.32. E. Hebey, M. Vaugon, Effective L p pinching for the concircular curvature, J. Geom. Anal. 6 (1996),531-553.33. Shaosai Huang, ǫ -regularity and structure of four-dimensional shrinking Ricci solitons, Int. Math. Res.Not. IMRN 2020, no. 5, 1511-1574.34. S. Ilias, Constantes explicites pour les in´egalit´es de Sobolev sur les vari´et´es riemanniennes compactes,Ann. Inst. Fourier (Grenoble) 33:2 (1983), 151-165.35. T. Ivey, Ricci solitons on compact three-manifolds, Diff. Geom. Appl. 3 (1993), 301-307.36. P. Li, S-T. Yau, On the Schr6dinger equation and the eigenvalue problem, Comm. Math. Phys. 88(1983), 309-318.37. H.-Z. Li, Y. Li, B. Wang, On the structure of Ricci shrinkers, arXiv:1809.04049v1.38. Y. Li, B. Wang, The rigidity of Ricci shrinkers of dimension four, Trans. Amer. Math. Soc. 371 (2019),6949-6972.39. Y. Li, B. Wang, Heat kernel on Ricci shrinkers, arXiv:1901.05691v1.40. E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schr¨odinger operators, Bull. Am. Math. Soc.82 (1976), 751-753.41. O. Munteanu, The volume growth of complete gradient shrinking Ricci solitons, arXiv:0904.0798v2.42. O. Munteanu, J.-P. Wang, Geometry of manifolds with densities, Adv. Math. 259 (2014), 269-305.43. O. Munteanu, J.-P. Wang, Positively curved shrinking Ricci solitons are compact, J. Diff. Geom. 106(2017), 499-505.44. L. Ni, N. Wallach, On a classification of gradient shrinking solitons, Math. Res. Lett. 15 (2008), 941-955.45. E.M. Ouhabaz, C. Poupaud, Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring estimates forSchr¨odinger operators on Riemannian manifolds, Acta Appl. Math. 110 (2010), 1449-1459.46. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002),arXiv:math.DG/0211159.47. P. Petersen, W. Wylie, On the classification of gradient Ricci solitons, Geom. Topol. 14 (2010), 2277-2300.48. S. Pigola, M. Rimoldi, A.G. Setti, Remarks on non-compact gradient Ricci solitons, Math. Z. 268 (2011),777-790.49. G.V. Rozenblum, The distribution of the discrete spectrum for singular differential operators. Sov.Math., Dokl. 13 (1972), 245-249; translation from Dokl. Akad. Nauk SSSR 202 (1972), 1012-1015.50. L. Saloff-Coste, Sobolev inequalities in familiar and unfamiliar settings, in Sobolev spaces in mathemat-ics, Vol I, 299-343, Int. Math. Ser. (N. Y.), 8, Springer, New York, 2009.51. M. Singer, Positive Einstein metrics with small L n/ -norm of the Weyl tensor, Differ. Geom. Appl. 2(1992), 269-274.52. H. Tran, On closed manifolds with harmonic Weyl curvature, Adv. Math. 322 (2017), 861-891.53. J.-Y. Wu, Elliptic gradient estimates for a weighted heat equation and applications, Math. Z. 280 (2015),451-468.54. J.-Y. Wu, Time analyticity for heat quation on gradient shrinking Ricci solitons, arXiv:1911.02735.55. J.-Y. Wu, Sharp Gaussian upper bounds for Schr¨odinger heat kernel on gradient shrinking Ricci solitons,arXiv 2006.13475. EOMETRIC INEQUALITIES AND RIGIDITY OF SHRINKERS 31
56. J.-Y. Wu, Sharp upper diameter bounds for compact shrinking Ricci solitons, arXiv 2008.02893.57. J.-Y. Wu, P. Wu, Heat kernel on smooth metric measure spaces with nonnegative curvature, Math.Ann., 362 (2015), 717-742.58. J.-Y. Wu, P. Wu, Heat kernel on smooth metric measure spaces and applications, Math. Ann. 365(2016), 309-344.59. J.-Y. Wu, P. Wu, Harmonic and Schrodinger functions of polynomial growth on gradient shrinking Riccisolitons, arXiv:1911.02729v4.60. P. Wu, A Weitzenbock formula for canonical metrics on four-manifolds, Trans. Amer. Math. Soc. 369(2017), 1079-1096.61. W. Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc.136 (2008), 1803-1806.62. Q.S. Zhang, Sobolev inequalities, heat kernels under Ricci flow, and the Poincar´e conjecture, CRC Press,Boca Raton, FL, 2011.63. Z.-H. Zhang, A gap theorem of four-dimensional gradient shrinking solitons, Comm. Anal. Geom. 28(2020), 729-742.
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