aa r X i v : . [ m a t h . DG ] S e p INTEGRAL PINCHED SHRINKING RICCI SOLITONS
GIOVANNI CATINO
Abstract.
We prove that a n -dimensional, 4 ≤ n ≤
6, compact gradient shrinking Riccisoliton satisfying a L n/ -pinching condition is isometric to a quotient of the round S n . Theproof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequalityand an improved rigidity result for integral pinched Einstein metrics. Key Words: Ricci solitons, Yamabe invariant, Integral pinched manifolds, Einstein metrics
AMS subject classification: 53C24, 53C25 Introduction
In this paper we investigate compact gradient shrinking Ricci solitons satisfying a L n/ -pinching condition. We recall that a Riemannian manifold ( M n , g ) is a gradient Ricci soliton if there exists a smooth function f on M n such that Ric + ∇ f = λg for some constant λ . The Ricci soliton is called shrinking if λ > steady if λ = 0 and expanding if λ <
0. The soliton is trivial if ∇ f is parallel. Ricci solitons generate self-similarsolutions of the Ricci flow, play a fundamental role in the formation of singularities and havebeen studied by several authors (see H.-D. Cao [5] for a nice overview).We will focus only on compact Ricci solitons. It has been shown by G. Perelman [24] thatthere is no non-trivial compact steady and expanding Ricci soliton. In dimension three, T.Ivey [18] proved that the only compact shrinking Ricci solitons are quotients of S with itsstandard metric. Dimension four is the lowest dimension where there are interesting examplesof shrinking Ricci solitons. The first examples where constructed by N. Koiso [20] and H.-D.Cao [4] (see also [10, 26]). Note that all of the known interesting compact shrinking Riccisolitons are K¨ahler.In the last years there have been a lot of interesting results concerning the classificationof compact shrinking Ricci solitons satisfying pointwise curvature conditions. For instance,it follows by the work of C. B¨ohm and B. Wilking [2] that the only compact shrinking Riccisolitons with positive (two-positive) curvature operator are quotients of S n . We have thesame conclusion also in the locally conformally flat case [9]. Other rigidity results on compactshrinkers can be found in [11, 21, 6, 8, 27, 7].In this paper we will show that quotients of S n are the only compact shrinking solitonssatisfying an integral curvature condition. We observe that all the results apply also tocomplete (possibly non-compact) gradient shrinking Ricci solitons with positive sectionalcurvature, since O. Munteanu and J. Wang in [22] recently showed that these conditions forcethe manifold to be compact. Date : September 25, 2015.
To fix the notation, we will denote by W , ◦ Ric and R the Weyl, traceless Ricci and scalarcurvature respectively. Y ( M, [ g ]) will be the Yamabe invariant associated to ( M n , g ) and thenorm of a ( k, l )-tensors T is defined as | T | g = g i m · · · g i k m k g j n . . . g j l n l T j ...j l i ...i k T n ...n l m ...m k . In dimension four we will prove the following L -pinching result. Theorem 1.1.
Every four-dimensional compact shrinking Ricci soliton satisfying Z M | W | dV g + Z M | ◦ Ric | dV g < Y ( M, [ g ]) is isometric to a quotient of the round S . As a corollary, using a lower bound for the Yamabe invariant proved in [13], we get thefollowing result.
Corollary 1.2.
Every four-dimensional compact shrinking Ricci soliton satisfying Z M | W | dV g + 54 Z M | ◦ Ric | dV g ≤ Z M R dV g is isometric to a quotient of the round S .Remark . The pinching condition in Corollary 1.2 is equivalent to the following (see Sec-tion 4) Z M | W | dV g + 239 Z M R dV g ≤ π χ ( M ) , where χ ( M ) is the Euler-Poincar´e characteristic of M .We notice that in Corollary 1.2 and Remark 1.3 we only have to assume the weak inequality.In fact, when equality occurs, we can show that ( M n , g ) has to be conformally Einstein, inparticular Bach-flat, and using [6] the conclusion follows (see Section 4).Theorem 1.1 is the four-dimensional case of the following result which holds in every di-mension 4 ≤ n ≤ Theorem 1.4.
Let ( M n , g ) be a n -dimensional, ≤ n ≤ , compact shrinking Ricci solitonssatisfying Z M (cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric (cid:13)∧ g (cid:12)(cid:12)(cid:12) n/ dV g ! n + s ( n − ( n − n − λV ( M ) n < s n − n − Y ( M, [ g ]) . Then ( M n , g ) is isometric to a quotient of the round S n . Moreover, in dimension ≤ n ≤ ,the same result holds only assuming the weak inequality. It is easy to see that, on a shrinking Ricci soliton of dimension n ≥
7, the pinching conditiondoes not hold, since λV ( M ) n = 1 n V ( M ) − nn Z M R dV g ≥ n Y ( M, [ g ]) . The proof of Theorem 1.4 is inspired by the classification of Einstein (or locally confor-mally flat) metrics satisfying a L n/ -pinching condition (see [16, 15, 13] and also [3]). Moreprecisely, we will use the soliton equation to obtain an elliptic PDE on | ◦ Ric | . Since everycompact shrinking soliton has positive scalar curvature, the positivity of the Yamabe invariant NTEGRAL PINCHED SHRINKING RICCI SOLITONS 3 Y ( M, [ g ]) implies a Sobolev-type inequality on | ◦ Ric | which, combined with the PDE, allowsus to get a L n/ -estimate on the curvature on every non-Einstein shrinking Ricci solitons. Indoing this, we will prove an algebraic curvature estimate (see Proposition 2.1) which holds inevery dimension and was firstly observed in dimension four in [3]. The pinching assumptionof Theorem 1.4 implies that the manifold ( M n , g ) has to be Einstein. To get the final result,one has to show that these Einstein metrics are actually of constant positive sectional curva-ture. It turns out that we cannot directly apply the result of E. Hebey and M. Vaugon [16,Theorem 1]. On the other hand, as already observed in their paper, one can improve theestimates using sharper algebraic inequalities on the Weyl curvature (see Lemma 2.3). Tothis aim, we will prove the following rigidity result for positively curved Einstein manifolds,which improves [16, Theorem 1] in dimension 4 ≤ n ≤ Theorem 1.5.
Let ( M n , g ) be a n -dimensional Einstein manifold with positive scalar curva-ture. There exists a positive constant A ( n ) such that if (cid:18)Z M | W | n dV g (cid:19) n < A ( n ) Y ( M, [ g ]) , then ( M n , g ) is isometric to a quotient of the round S n .We can take A (4) = √ , A (5) = , A (6) = √ √ , A ( n ) = n − n − if ≤ n ≤ and A ( n ) = n if n ≥ . Algebraic preliminaries
To fix the notation we recall that the Riemann curvature operator of a Riemannian manifold( M n , g ) is defined as in [12] by Rm ( X, Y ) Z = ∇ Y ∇ X Z − ∇ X ∇ Y Z + ∇ [ X,Y ] Z .
In a local coordinate system the components of the (3 , R lijk ∂∂x l = Rm (cid:0) ∂∂x i , ∂∂x j (cid:1) ∂∂x k and we denote by R ijkl = g lp R pijk its (4 , Ric is obtained by the contraction (
Ric ) ik = R ik = g jl R ijkl , R = g ik R ik will denote the scalar curvature and ( ◦ Ric ) ik = R ik − n R g ik the traceless Riccitensor. The Weyl tensor W is then defined by the following orthogonal decomposition formula(see [12, Chapter 3, Section K]) in dimension n ≥ W = Rm − n − ◦ Ric (cid:13)∧ g − n ( n − R g (cid:13)∧ g where (cid:13)∧ denotes the Kulkarni-Nomizu product which is defined as follow: let A, B twosymmetric (0 , A (cid:13)∧ B ) ijkl = A ik B jl − A il B jk − A jk B il + A jl B ik . The Riemannian metric induces norms on all the tensor bundles, in coordinates this norm isgiven, for a tensor T = T j ...j l i ...i k , by | T | g = g i m · · · g i k m k g j n . . . g j l n l T j ...j l i ...i k T n ...n l m ...m k . In order to prove Theorem 1.4, we will use two algebraic curvature inequalities. The firstone, which involves the Weyl curvature and the traceless-Ricci curvature can be viewed as a
GIOVANNI CATINO combination of the algebraic inequality for trace-free symmetric two-tensors T = { T ij } (for aproof see [17, Lemma 2.4]) | T ij T jk T ik | ≤ s n − n ( n − | T | (2.1)with Huisken inequality [17, Lemma 3.4] (cid:12)(cid:12)(cid:12)(cid:12) W ijkl ◦ R ik ◦ R jl (cid:12)(cid:12)(cid:12)(cid:12) ≤ s n − n − | W || ◦ Ric | . (2.2)The following estimates was proved in dimension four in [3, Lemma 4.7]. Proposition 2.1.
On every n -dimensional Riemannian manifold the following estimate holds (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 − (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) / | ◦ Ric | . Proof.
We follow [3, Lemma 4.7]. First of all we have( ◦ Ric (cid:13)∧ g ) ijkl = ( ◦ R ik g jl − ◦ R il g jk + ◦ R jl g ik − ◦ R jk g il )( ◦ Ric (cid:13)∧ ◦ Ric ) ijkl = 2( ◦ R ik ◦ R jl − ◦ R il ◦ R jk ) . An easy computation shows W ijkl ◦ R ik ◦ R jl = 14 W ijkl ( ◦ Ric (cid:13)∧ ◦ Ric ) ijkl ◦ R ij ◦ R jk ◦ R ik = −
18 ( ◦ Ric (cid:13)∧ g ) ijkl ( ◦ Ric (cid:13)∧ ◦ Ric ) ijkl . Hence we get the following identity − W ijkl ◦ R ik ◦ R jl + 2 n − ◦ R ij ◦ R jk ◦ R ik = − (cid:18) W + 1 n − ◦ Ric (cid:13)∧ g (cid:19) ijkl ( ◦ Ric (cid:13)∧ ◦ Ric ) ijkl . (2.3)Since ◦ Ric (cid:13)∧ ◦ Ric has the same symmetries of the Riemann tensor, it can be orthogonallydecomposed as ◦ Ric (cid:13)∧ ◦ Ric = T + V + U where T is totally trace-free and V ijkl = − n − ◦ Ric (cid:13)∧ g ) ijkl + 2 n ( n − | ◦ Ric | ( g (cid:13)∧ g ) ijkl U ijkl = − n ( n − | ◦ Ric | ( g (cid:13)∧ g ) ijkl , where ( ◦ Ric ) ik = ◦ R ip ◦ R kp . Taking the squared norm one obtains | ◦ Ric (cid:13)∧ ◦ Ric | = 8 | ◦ Ric | − | ◦ Ric | | V | = 16 n − | ◦ Ric | − n ( n − | ◦ Ric | | U | = 8 n ( n − | ◦ Ric | . NTEGRAL PINCHED SHRINKING RICCI SOLITONS 5
In particular, one has | T | + n | V | = | ◦ Ric (cid:13)∧ ◦ Ric | + n − | V | − | U | = 8( n − n − | ◦ Ric | . We now estimate the right hand side of (2.3). Using the fact that W and T are totallytrace-free and the Cauchy-Schwarz inequality we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) W + 1 n − ◦ Ric (cid:13)∧ g (cid:19) ijkl ( ◦ Ric (cid:13)∧ ◦ Ric ) ijkl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) W + 1 n − ◦ Ric (cid:13)∧ g (cid:19) ijkl ( T + V ) ijkl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric (cid:13)∧ g ! ijkl (cid:18) T + r n V (cid:19) ijkl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric (cid:13)∧ g !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) | T | + n | V | (cid:17) = 8( n − n − (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) | ◦ Ric | . This estimate together with (2.3) concludes the proof. (cid:3)
Remark . Notice that this estimate is stronger than the one obtained by using triangleinequality together with inequalities (2.1) and (2.2).We conclude this section with a second algebraic inequality on the Weyl tensor. Let T = { T ijkl } be a tensor with the same symmetries as the Riemann tensor. It defines a symmetricoperator T : Λ ( M ) −→ Λ ( M ) on the space of two-forms by( T ω ) kl := 12 T ijkl ω ij , with ω ∈ Λ ( M ). Hence we have that µ is an eigenvalue of T if T ijkl ω ij = 2 µ ω kl , for some0 = ω ∈ Λ ( M ) and we have k T k = | T | . Lemma 2.3.
On every n -dimensional Riemannian manifold there exists a positive constant C ( n ) such that the following estimate holds 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 ≥ .Proof. First of all, simply by Cauchy-Schwartz, in every dimension one has2 W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij ≤ | W | . In dimension n = 4, the sharper constant C (4) = √ was proved in [17, Lemma 3.5].To get the estimate in dimension n = 5, we invoke the following algebraic identity whichholds on every five-dimensional Riemannian manifolds (for instance see [19, Equation (A.3)]) W ijkl W klpq W pqij = 4 W ijkl W ipkq W pjql . GIOVANNI CATINO
Hence, by Cauchy-Schwartz, in dimension n = 5 one has2 W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij = W ijkl W klpq W pqij ≤ | W | . Finally, to obtain the estimate in dimension n = 6 we proceed as in [17] using an idea ofS. Tachibana [25] and define for some fixed indices p, q, r, s the local skew symmetric tensor ω = { ω ( pqrs ) ij } by ω ( pqrs ) ij := W iqrs g jp + W pirs g jq + W pqis g jr + W pqri g js − W jqrs g ip − W pjrs g iq − W pqjs g ir − W pqrj g is . Then a simple computation shows that2 W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij = 116 W ijkl ω ( pqrs ) ij ω ( pqrs ) kl (2.4) | ω | = 8( n − | W | . Now if we denote by µ the maximum eigenvalue of W , since W is trace-free, it follows from [17,Corollary 2.5] that W ijkl ω ( pqrs ) ij ω ( pqrs ) kl ≤ µ | ω | ≤ s ( n − n + 1) n ( n − | W || ω | = 8( n − s ( n − n + 1) n ( n − | W | and we obtain2 W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij ≤ r ( n − n − n + 1) n | W | . Substituting in dimension n = 6, we get C (6) = √ √ < . (cid:3) Proof of the main theorem
In this section we will prove Theorem 1.4. Let ( M n , g ) be a n -dimensional compact gradientshrinking Ricci solitons Ric + ∇ f = λg for some smooth function f and some positive constant λ >
0. First of all we recall thefollowing well known formulas (for the proof see [9])
Lemma 3.1.
Let ( M n , g ) be a gradient Ricci soliton, then the following formulas hold ∆ f = nλ − R (3.1)∆ f R = 2 λR − | Ric | (3.2)∆ f R ik = 2 λR ik − W ijkl R jl (3.3)+ 2( n − n − (cid:0) R g ik − nR R ik + 2( n − R ij R jk − ( n − | Ric | g ik (cid:1) , where the ∆ f denotes the f -Laplacian, ∆ f = ∆ − ∇ ∇ f . NTEGRAL PINCHED SHRINKING RICCI SOLITONS 7
In particular, using ◦ R ij = R ij − n Rg ij and the fact that R ij R jk R ik = ◦ R ij ◦ R jk ◦ R ik + 3 n R | Ric | − n R a simple computation shows the following equation for the f -Laplacian of the squared normof the treceless Ricci tensor Lemma 3.2.
Let ( M n , g ) be a gradient Ricci soliton, then the following formula holds
12 ∆ f | ◦ Ric | = |∇ ◦ Ric | + 2 λ | ◦ Ric | − W ijkl ◦ R ik ◦ R jl + 4 n − ◦ R ij ◦ R jk ◦ R ik − n − n ( n − R | ◦ Ric | . Moreover, it follows immediately from equation (3.2) and the maximum principle that everycompact shrinking solitons has positive scalar curvature (see [18]).We will denote Y ( M, [ g ]) the Yamabe invariant associated to ( M n , g ) (here [ g ] is the con-formal class of g ) defined by Y ( M, [ g ]) = inf e g ∈ [ g ] R M e R dV e g (cid:0)R M dV e g (cid:1) ( n − /n = 4( n − n − u ∈ W , ( M ) R M |∇ u | dV g + n − n − R M R u dV g (cid:0)R M | u | n/ ( n − dV g (cid:1) ( n − /n It is well known that, on a compact manifold, Y ( M, [ g ]) is positive (respectively zero or nega-tive) if and only if there exists a conformal metric in [ g ] with everywhere positive (respectivelyzero or negative) scalar curvature. Then, every compact shrinking soliton has positive Yamabeinvariant Y ( M, [ g ]) >
0, so for every u ∈ W , ( M ) the following Yamabe-Sobolev inequalityholds n − n − Y ( M, [ g ]) (cid:18)Z M | u | nn − dV g (cid:19) n − n ≤ Z M |∇ u | dV g + n − n − Z M R u dV g . (3.4)Now, let us assume that 4 ≤ n ≤ M n , g ) satisfies the integral pinching condition asin Theorem 1.4 Z M (cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric (cid:13)∧ g (cid:12)(cid:12)(cid:12) n/ dV g ! n + s ( n − ( n − n − λV ( M ) n < s n − n − Y ( M, [ g ]) , (3.5)with V ( M ) := R M dV g . From Lemma 3.2 we have12 ∆ f | ◦ Ric | = |∇ ◦ Ric | + 2 λ | ◦ Ric | − W ijkl ◦ R ik ◦ R jl + 4 n − ◦ R ij ◦ R jk ◦ R ik − n − n ( n − R | ◦ Ric | . Using Kato inequality ( |∇ ◦ Ric | ≥ |∇| ◦ Ric || at every point where | ◦ Ric | 6 = 0) and Proposi-tion 2.1 we obtain0 ≥ −
12 ∆ f | ◦ Ric | + |∇| ◦ Ric || + 2 λ | ◦ Ric | − r n − n − (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) / | ◦ Ric | − n − n ( n − R | ◦ Ric | . GIOVANNI CATINO
Integrating by parts over M n and using equation (3.1) it follows that0 ≥ − Z M | ◦ Ric | ∆ f dV g + Z M |∇| ◦ Ric || dV g + 2 λ Z M | ◦ Ric | dV g − r n − n − Z M (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) / | ◦ Ric | dV g − n − n ( n − Z M R | ◦ Ric | dV g = Z M |∇| ◦ Ric || dV g − n − λ Z M | ◦ Ric | dV g − r n − n − Z M (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) / | ◦ Ric | dV g + n − n + 82 n ( n − Z M R | ◦ Ric | dV g Using inequality (3.4) with u := | ◦ Ric | we get0 ≥ n − n − Y ( M, [ g ]) (cid:18)Z M | ◦ Ric | nn − dV g (cid:19) n − n − n − λ Z M | ◦ Ric | dV g − r n − n − Z M (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) / | ◦ Ric | dV g + ( n − n ( n − Z M R | ◦ Ric | dV g From H¨older inequality, since λ > n ≥
4, we obtain0 ≥ n − n − Y ( M, [ g ]) − n − λV ( M ) n − r n − n − Z M (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) n/ dV g ! n (cid:18)Z M | ◦ Ric | nn − dV g (cid:19) n − n + ( n − n ( n − Z M R | ◦ Ric | dV g . Thus, either | ◦ Ric | ≡
0, i.e. ( M n , g ) is Einstein, or the following estimate holds Z M (cid:18) | W | + 8 n ( n − | ◦ Ric | (cid:19) n/ dV g ! n + s ( n − ( n − n − λV ( M ) n ≥ s n − n − Y ( M, [ g ]) . Since W is totally trace-free, one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W + √ √ n ( n − ◦ Ric (cid:13)∧ g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | W | + 8 n ( n − | ◦ Ric | and the pinching condition (3.5) implies that ( M n , g ) is Einstein. Moreover, observe that indimension n = 5 ,
6, we get the same conclusion if we assume just the weak inequality in (3.5).Since g is Einstein, by Obata Theorem [23] one has that g is a (the) Yamabe metric of [ g ],i.e. Y ( M, [ g ]) = V ( M ) − nn Z M R dV g . Moreover, integrating equation (3.1) one has λV ( M ) n = 1 n V ( M ) − nn Z M R dV g = 1 n Y ( M, [ g ]) . NTEGRAL PINCHED SHRINKING RICCI SOLITONS 9
Hence, the pinching condition (3.5) implies (cid:18)Z M | W | n dV g (cid:19) n ≤ n − n − n p n − n − Y ( M, [ g ]) . (3.6)To conclude the proof of Theorem 1.4 we need to show that an Einstein manifold of dimension4 ≤ n ≤ ≤ n ≤
6. The next result is Theorem 1.5 in the introduction.
Theorem 3.3.
Let ( M n , g ) be a n -dimensional Einstein manifold with positive scalar curva-ture. There exists a positive constant A ( n ) such that if (cid:18)Z M | W | n dV g (cid:19) n < A ( n ) Y ( M, [ g ]) , then ( M n , g ) is isometric to a quotient of the round S n .We can take A (4) = √ , A (5) = , A (6) = √ √ , A ( n ) = n − n − if ≤ n ≤ and A ( n ) = n if n ≥ .Proof. Following the proof in [16], by Bianchi identities and the fact that g is Einstein, onecan prove that the Weyl tensor satisfies the following equation (see [16, Equation (5)])∆ W ijkl = 2 n R W ijkl − (cid:18) W ipkq W pjql + 12 W klpq W pqij (cid:19) . Contracting with W ijkl , we get12 ∆ | W | = |∇ W | + 2 n R | W | − (cid:18) W ijkl W ipkq W pjql + 12 W ijkl W klpq W pqij (cid:19) . Integrating on M n , using again Kato inequality and Lemma 2.3, one obtains0 ≥ Z M |∇| W || dV g + 2 n Z M R | W | dV g − C ( n ) Z M | W | dV g , where C ( n ) is defined as in Lemma 2.3. Using H¨older inequality and (3.4) with u := | W | , weget0 ≥ Z M |∇| W || dV g + 2 n Z M R | W | dV g − C ( n ) (cid:18)Z M | W | n dV g (cid:19) n (cid:18)Z M | W | nn − dV g (cid:19) n − n ≥ Z M |∇| W || dV g + 2 n Z M R | W | dV g − C ( n ) Y ( M, [ g ]) (cid:18)Z M | W | n dV g (cid:19) n (cid:18) n − n − Z M |∇| W || dV g + Z M R | W | dV g (cid:19) Now by assumption one has (cid:18)Z M | W | n dV g (cid:19) n < A ( n ) Y ( M, [ g ]) . It then follows that | W | ≡
0, so g has constant positive sectional curvature, if A ( n ) satisfiesthe following inequalities ( C ( n ) A ( n ) ≤ n − n − C ( n ) A ( n ) ≤ n . Since C (5) = 1, C (6) = √ √ and C ( n ) = for n ≥
7, it is easy to see that we can take, A (5) = , A (6) = √ √ , A ( n ) = n − n − if 7 ≤ n ≤ A ( n ) = n if n ≥
10. Thisconcludes the proof of the theorem when n = 4.In dimension n = 4 we can improve the estimate by using the following refined Katoinequality proved in [14] (see also [3, Lemma 4.3]) which holds on every four-dimensionalRiemannian manifold with harmonic Weyl tensor (in particular if g is Einstein) |∇ W | ≥ |∇| W || . Hence reasoning as before, we have0 ≥ Z M |∇| W || dV g + 12 Z M R | W | dV g − C (4) Y ( M, [ g ]) (cid:18)Z M | W | dV g (cid:19) (cid:18) Z M |∇| W || dV g + Z M R | W | dV g (cid:19) Hence, we need ( C (4) A (4) ≤ C (4) A (4) ≤ . Since C (4) = √ we obtain A (4) = √ . (cid:3) Remark . As already observed in the introduction, this result was proved in [16] with aslightly stronger pinching assumption in dimension 4 ≤ n ≤
6. In particular, in dimension n = 5, their theorem does not apply under condition (3.6).4. Four dimensional shrinking Ricci solitons
In this section we will prove Corollary 1.2 and Remark 1.3. Let ( M , g ) be a compact four-dimensional Riemannian manifold. First of all we recall the Chern-Gauss-Bonnet formula(see [1, Equation 6.31]) Z M | W | dV g − Z M | ◦ Ric | dV g + 16 Z M R dV g = 32 π χ ( M ) . (4.1)If we denote with σ ( A ) the second-elementary function of the eigenvalues of the Schoutentensor A := (cid:0) Ric − R g (cid:1) , it is easy to see that σ ( A ) = 196 R − | ◦ Ric | and the Chern-Gauss-Bonnet formula reads Z M | W | dV g + 16 Z M σ ( A ) dV g = 32 π χ ( M ) . NTEGRAL PINCHED SHRINKING RICCI SOLITONS 11
By the conformal invariance of the L -norm of the Weyl tensor in dimension four, it followsthat the integral of σ ( A ) is conformally invariant too. To prove Corollary 1.2, we need thefollowing lower bound for the Yamabe invariant which was proved by M. J. Gursky [13]. Sincethe proof is short, for the sake of completeness, we include it. Lemma 4.1.
Let ( M , g ) be a compact four-dimensional manifold. Then, the following esti-mate holds Y ( M, [ g ]) ≥ Z M σ ( A ) dV g = Z M R dV g − Z M | ◦ Ric | dV g . Moreover, the inequality is strict unless ( M , g ) is conformally Einstein.Proof. We solve the Yamabe problem in [ g ]. Let e g ∈ [ g ] be a Yamabe metric. Then Y ( M, [ g ]) = (cid:16)R M e R dV e g (cid:17) R M dV e g = Z M e R dV e g ≥ Z M e R dV e g − Z M | g ◦ Ric | dV e g = 96 Z M σ ( e A ) dV e g = 96 Z M σ ( A ) dV g , where in the last equality we have used the conformally invariance of R M σ ( A ) dV g . Theequality case follows immediately. (cid:3) From this lemma we get Z M | W | dV g + Z M | ◦ Ric | dV g − Y ( M, [ g ]) ≤ Z M | W | dV g + 54 Z M | ◦ Ric | dV g − Z M R dV g . Moreover, the inequality is strict unless ( M , g ) is conformally Einstein. In the first case,Theorem 1.1 immediately implies Corollary 1.2. In the second case, the fact that g is con-formally Einstein, in particular, implies that ( M , g ) is Bach flat (see [1, Proposition 4.78]).Since M is compact, by a result of H.-D. Cao and Q. Chen [6] it follows that ( M , g ) hasto be Einstein. Following the proof of Theorem 1.4, we can see that the pinching conditiontogether with Theorem 3.3 concludes the proof of Corollary 1.2.Finally, observe that by the Chern-Gauss-Bonnet formula (4.1), the right-hand side can bewritten as Z M | W | dV g + 54 Z M | ◦ Ric | dV g − Z M R dV g = 138 Z M | W | dV g + 112 Z M R dV g − π χ ( M ) . This proves Remark 1.3.
Acknowledgments .
The author is member of the Gruppo Nazionale per l’Analisi Matemat-ica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matem-atica (INdAM).
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NTEGRAL PINCHED SHRINKING RICCI SOLITONS 13 (Giovanni Catino)
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