aa r X i v : . [ m a t h . DG ] F e b DEFORMATIONS OF Q -CURVATURE II YUEH-JU LIN AND WEI YUAN
Abstract.
This is the second article of a sequence of research on deformations of Q -curvature. In the previous one, we studied local stability and rigidity phenomena of Q -curvature. In this article, we mainly investigate the volume comparison with respect to Q -curvature. In particular, we show that volume comparison theorem holds for metricsclose to strictly stable positive Einstein metrics. This result shows that Q -curvature canstill control the volume of manifolds under certain conditions, which provides a fundamentalgeometric characterization of Q -curvature. Applying the same technique, we derive the localrigidity of strictly stable Ricci-flat manifolds with respect to Q -curvature, which shows thenon-existence of metrics with positive Q -curvature near the reference metric. Introduction
The Q -curvature is a th -order scalar type curvature. It has been studied for decades dueto its geometric resemblance to Gaussian and scalar curvature as a higher-order curvaturequantity.For a closed -dimensional Riemannian manifold ( M , g ) , Q -curvature is defined to be Q g = −
16 ∆ g R g − | Ric g | g + 16 R g . (1.1)It satisfies the Gauss-Bonnet-Chern Formula Z M (cid:18) Q g + 14 | W g | g (cid:19) dv g = 8 π χ ( M ) , where R g , Ric g , and W g are scalar curvature, Ricci curvature, and Weyl tensor for ( M , g ) respectively. In particular, if ( M , g ) is locally conformally flat, i.e. W g = 0 , it reduces to Z M Q g dv g = 8 π χ ( M ) , which can be viewed as a generalization of the classic Gauss-Bonnet Theorem for closedsurfaces.Branson ([2]) extended (1.1) and defined the Q-curvature for manifolds with dimension atleast three to be Q g = A n ∆ g R g + B n | Ric g | g + C n R g , (1.2) Key words and phrases. Q -curvature, volume comparison, Einstein manifolds.Wei Yuan was supported by NSFC (Grant No. 12071489, No. 11521101). here A n = − n − , B n = − n − and C n = n ( n − n − n − ( n − . With the aid of the Paneitzoperator ([17]) P g = ∆ g − div g [( a n R g g + b n Ric g ) d ] + n − Q g , where a n = ( n − +42( n − n − and b n = − n − , Q -curvature shares a similar conformal transforma-tion law as scalar curvature: Q ˆ g = e − u ( P g u + Q g ) , for n = 4 and ˆ g = e u gQ ˆ g = 2 n − u − n +4 n − P g u, for n = 4 and ˆ g = u n − g This suggests that Q-curvature is an th -order analogue of scalar curvature.For a long time, mathematicians are seeking for a better understanding about geometricinterpretations of Q-curvature especially in dimensions five and above. In the field of con-formal geometry, there has been many excellent works regarding Q -curvature (see [9] for agreat survey). Without the restrictions in conformal classes, there are not many results on Q -curvature from the viewpoint of Riemannian geometry so far.The main purpose of the authors’ research on Q -curvature is to investigate the Riemanniangeometric properties of Q -curvature. For instances, the authors studied local stability andrigidity phenomena and derived some interesting geometric results about Q -curvature (see[13, 14] for more details). These results strongly suggest that Q -curvature shares analogousgeometric properties as scalar curvature.Volume comparison theorem is a fundamental result in differential geometry. It is impor-tant both theoretically and practically in the analysis of geometric problems. The classicvolume comparison states that a lower bound for Ricci curvature implies the volume compar-ison of geodesic balls with those in the model spaces. A natural question is that whether wecan replace the assumption on Ricci curvature by a weaker one? As for the scalar curvature,this idea has been proved to be feasible in some special situations ([20]).Inspired by the second author’s work [20], we consider the volume comparison for Q-curvature. As the first step, we give the definition of model spaces: Definition 1.1.
A Riemannian manifold ( M n , ¯ g ) is Q -critical , if there is a nontrivial function f ∈ C ∞ ( M ) and a constant κ ∈ R such that Γ ∗ ¯ g f = κ ¯ g, where Γ ∗ ¯ g : C ∞ ( M ) → S ( M ) is the L -formal adjoint of Γ ¯ g := DQ ¯ g , the linearization of Q -curvature at ¯ g .The concept of Q -critical metrics provides a standard model for volume comparison of Q -curvature. That is, we only need to consider the volume comparison with respect to Q -critical metrics. The reason is that for non- Q -critical metrics, one can perturb Q -curvatureand the volume simultaneously without any constraint: heorem 1.2 (Case-Lin-Yuan [4]) . Let ( M n , g ) be a closed Riemannian manifold. If ( M n , g ) is not Q -critical, then there are neighborhoods U of Riemannian metrics of g and V ⊂ C ∞ ( M ) ⊕ R of ( Q g , V M ( g )) such that for any ( ψ, υ ) ∈ V , there is a metric ˆ g ∈ U such that Q ˆ g = ψ and V M (ˆ g ) = υ .Remark . Corvino, Eichmair, and Miao first showed such type of theorem holds for scalarcurvature ([6]). For a more general version regarding conformally variational invariants ,please refer to [4].As a special case of Q -critical metrics, J -Einstein metrics are the most basic examples([14]): Definition 1.4.
Let ( M n , g ) be a Riemannian manifold ( n ≥ . We define a symmetric -tensor associated to Q -curvature called J -tensor to be J g := −
12 Γ ∗ g (1) . A metric is called J -Einstein , if J g = Λ g for some constant Λ ∈ R .In particular, if ¯ g is Einstein, one can check that J ¯ g = 1 n Q ¯ g ¯ g = ( n − n + 2)8 n ( n − R g ¯ g. Therefore, Einstein metrics are J -Einstein and hence Q -critical. However, under certainconditions, a J -Einstein metric has to be Einstein. We present a characterization of Einsteinmetrics in terms of the spectrum of the Einstein operator ∆ ¯ gE = ∆ ¯ g + 2 Rm ¯ g defined on the space of symmetric -tensors. Theorem 1.5.
Suppose ( M n , ¯ g ) is an n -dimensional closed J -Einstein manifold and theEinstein operator ∆ ¯ gE on S ( M ) satisfies Λ ¯ gE := inf h ∈ S ( M ) \{ } R M h h, − ∆ ¯ gE h i ¯ g dv ¯ g R M | h | g dv ¯ g > − ( n − ( n + 2)8 n ( n − min M R ¯ g . Furthermore, we assume the scalar curvature R ¯ g is a constant when ≤ n ≤ . Then ¯ g isan Einstein metric. This result is an important motivation for one to take Einstein metrics to be reference met-rics when considering the volume comparison with respect to Q -curvature. Unfortunately,being an Einstein metric is not sufficient for volume comparison to hold. In fact, one needsto impose a stronger assumption on Einstein metrics. Definition 1.6 (Stability of Einstein manifolds [1, 10]) . For n ≥ , suppose ( M n , ¯ g ) is aclosed Einstein manifold. The Einstein metric ¯ g is said to be strictly stable , if the Einsteinoperator ∆ ¯ gE = ∆ ¯ g + 2 Rm ¯ g s a negative operator on S TT , ¯ g ( M ) \{ } , where S TT , ¯ g ( M ) := { h ∈ S ( M ) | δ ¯ g h = 0 , tr ¯ g h = 0 } is the space of transverse-traceless symmetric 2-tensors on ( M n , ¯ g ) .Now we state our main result in this article, which concerns a volume comparison withrespect to Q -curvature for closed strictly stable Einstein manifolds. Theorem 1.7.
For n ≥ , suppose ( M n , ¯ g ) is an n -dimensional closed strictly stable Einsteinmanifold with Ricci curvature Ric ¯ g = ( n − λ ¯ g, where λ > is a constant. Then there exists a constant ε > such that for any metric g on M satisfying Q g ≥ Q ¯ g and || g − ¯ g || C ( M, ¯ g ) < ε , the following volume comparison holds V M ( g ) ≤ V M (¯ g ) , with the equality holds if and only if g is isometric to ¯ g .Remark . The above volume comparison does not hold for Ricci flat metrics. This is easyto see by taking g = c ¯ g for some constant c = 0 . Clearly, the Q -curvature is Q g = Q ¯ g = 0 ,but the volume V M ( g ) can be either larger or smaller than V M (¯ g ) depending on c > or c < . Remark . The strictly stability condition in Theorem 1.7 is the same as the scalar cur-vature case (see [20]). It is also necessary for Q -curvature. See Remark 6.1 for more detailsregarding a counterexample.In particular, Theorem 1.7 implies the volume comparison for metrics near the sphericalmetric, since the reference metric is strictly stable. This is a th -order analogue of Bray’sconjecture for scalar curvature (see [20] for more details).
Corollary 1.10.
For n ≥ , let ( S n , ¯ g ) be the canonical sphere with Ricci curvature Ric ¯ g = ( n − g. Then there exists a constant ε > such that for any metric g on S n satisfying Q g ≥ n ( n − n + 2) and || g − ¯ g || C ( M, ¯ g ) < ε , the following volume comparison holds V M ( g ) ≤ V S n , with the equality holds if and only if g is isometric to ¯ g . ccording to Remark 1.8, the volume comparison for Ricci-flat manifolds can not beexpected. However, applying the same idea as proof of Theorem 1.7, we can show thatstrictly stable Ricci-flat manifolds admits local rigidity with respect to Q -curvature. Thisextends our previous local rigidity result for tori and answers the question proposed by thereferee of our earlier article [13]. Theorem 1.11.
Suppose ( M n , ¯ g ) is a strictly stable Ricci-flat manifold, then there exists aconstant ε > such that any metric g satisfying Q g ≥ and || g − ¯ g || C ( M, ¯ g ) < ε implies g has to be Ricci-flat. In particular, there is no metric with positive Q -curvaturenear ¯ g .Remark . It is not difficult to improve this local rigidity result by a weaker assumptionthat the Ricci-flat metric ¯ g is only stable instead. It would be interesting to ask whetherwe can find an example of unstable Ricci-flat manifold which admits a metric of positiveQ-curvature.This article is organized as follows: In Section 2, we introduce the notation and usefulformulas needed throughout the article. In Section 3, we show the rigidity of Einstein met-rics in the category of J -Einstein metrics and calculate the first variation of J -tensor atEinstein metrics which will be used in Section 5. In Section 4, we calculate the variationalformulas for the main functional. In Section 5, we prove our main results (Theorem 1.7,1.11) by showing nonpositivity of second variation of the functional and using Morse lemmaargument. In Section 6, we provide a counterexample showing that the strictly stability ofEinstein metric is necessary for our main result. We also make some observations about aglobal volume comparison of Q -curvature for a locally conformally flat -manifold. Acknowledgement . The authors would like to express their appreciations to ProfessorsJeffrey S. Case and Yen-Chang Huang for many inspiring discussions. We would like tothank Professor Yoshihiko Matsumoto for introducing his remarkable work [15] and Profes-sor Mijia Lai for a valuable comment on Corollary 6.3. Yueh-Ju Lin would also like to thankPrinceton University for the support, as part of the work was done when she was in Princeton.2.
Preliminary and notation
Notations.
Throughout this article, we will always assume ( M n , g ) to be an n -dimensionalclosed Riemannian manifold ( n ≥ ) unless otherwise stated. Also, we list notations involvedin this article: M - the set of all smooth metrics on M ; ( M ) - the set of all smooth diffeomorphisms ϕ : M → M ; X ( M ) - the set of all smooth vector fields on M ; S ( M ) - the set of all smooth symmetric 2-tensors on M ; V M ( g ) - the volume of manifold M with respect to the metric g .We adopt the following convention for Ricci curvature tensor R jk = R iijk = g il R ijkl . and denote its traceless part as E g := Ric g − n R g g. The Schouten tensor is defined to be S g = 1 n − (cid:18) Ric g − n − R g g (cid:19) and ˚ S g is denoted to be its traceless part.For Laplacian operator, we use the convention as follows ∆ g := g ij ∇ i ∇ j . For simplicity, we introduce following operations: ( h × k ) ij := g kl h ik k jl = h li k lj , h · k := tr g ( h × k ) = g ij g kl h ik k jl = h jk k jk and ( Rm · h ) jk := R ijkl h il for any h, k ∈ S ( M ) .Let X ∈ X ( M ) and h ∈ S ( M ) , we use following notations for the operator ( δ g h ) i := − ( div g h ) i = −∇ j h ij , which is the L -formal adjoint of Lie derivative (up to a scalar multiple)
12 ( L g X ) ij = 12 ( ∇ i X j + ∇ j X i ) . The Einstein operator acting on h ∈ S ( M ) is defined to be ∆ gE h = ∆ g h + 2 Rm g · h. The J -tensor ([14]) is defined to be J g = 1 n Q g g − n − B g − n − n − n − T g , where B g =∆ gE ˚ S g − ∇ g ( tr g S g ) + 1 n g ∆ g ( tr g S g ) − ( n − S g − | ˚ S g | g g − n − n ( tr g S g )˚ S g s the Bach tensor and T g :=( n − (cid:18) ∇ g ( tr g S g ) − n g ∆ g ( tr g S g ) (cid:19) + 4( n − (cid:18) ˚ S g − n | ˚ S g | g g (cid:19) − ( n − n + 2 n − n ( tr g S g )˚ S g . Basic variational formulae.
We list several formulas for linearization of geometricquantities that will be useful for later sections (see [7, 13, 20] for detailed calculations).The linearization of Ricci tensor is ( DRic g ) · h = − (cid:2) ∆ gE h − ( Ric g × h + h × Ric g ) + ∇ g ( tr g h ) + ( ∇ j ( δ g h ) k + ∇ k ( δ g h ) j ) dx j ⊗ dx k (cid:3) , and the linearization of scalar curvature is ( DR g ) · h = − ∆ g ( tr g h ) + δ g h − Ric g · h. The linearization of Q -curvature is Γ g h :=( DQ g ) · h = A n (cid:20) − ∆ g ( tr g h ) + ∆ g δ g h − ∆ g ( Ric g · h ) + 12 dR g · ( d ( tr g h ) + 2 δ g h ) − ∇ g R g · h (cid:21) − B n (cid:2) Ric g · ∆ gE h + Ric g · ∇ g ( tr g h ) + 2 Ric g · ∇ ( δ g h ) (cid:3) + 2 C n R g (cid:2) − ∆ g ( tr g h ) + δ g h − Ric g · h (cid:3) . and the L -formal adjoint of Γ g is Γ ∗ g f := A n (cid:20) ∇ g ∆ g f − g ∆ g f − Ric g ∆ g f + 12 gδ g ( f dR g ) + ∇ g ( f dR g ) − f ∇ g R g (cid:21) − B n (cid:2) ∆ g ( f Ric g ) + 2 f ( Rm g · Ric g ) + gδ g ( f Ric g ) + 2 ∇ g δ g ( f Ric g ) (cid:3) − C n (cid:2) g ∆ g ( f R g ) − ∇ g ( f R g ) + f R g Ric g (cid:3) , where A n , B n , C n are defined in (1.2). The first and second variations of the volume functionalare ( D V M,g ) · h = 12 Z M ( tr g h ) dv g and ( D V M,g ) · ( h, h ) = 14 Z M [( tr g h ) − | h | g ] dv g . J -tensor and Einstein metrics In this section, we will discuss some involved topics about J -tensor and Einstein metrics. .1. Rigidity of Einstein metrics in the category of J -Einstein metrics. As we have stated in the introduction, Einstein metrics can be identified with a charac-terization in the spectrum of Einstein operator in the category of J -Einstein metrics. Nowwe present a simple proof here. Proof of Theorem 1.5.
By definition, the J -Einstein metric ¯ g satisfies the equation ˚ J ¯ g = − n − (cid:18) B ¯ g + n − n − T ¯ g (cid:19) = − n − (cid:20) ∆ ¯ gE ˚ S ¯ g + n − n + 124( n − (cid:18) ∇ g ( tr ¯ g S ¯ g ) − n ¯ g ∆ ¯ g ( tr ¯ g S ¯ g ) (cid:19)(cid:21) + 2 n | ˚ S ¯ g | g ¯ g + ( n − ( n + 2)4 n ( n −
1) ( tr ¯ g S ¯ g )˚ S ¯ g =0 . It implies Z M h ˚ J ¯ g , ˚ S ¯ g i ¯ g dv ¯ g = 1 n − Z M (cid:20) − D ˚ S ¯ g , ∆ ¯ gE ˚ S ¯ g E ¯ g + n − n + 124 n | d ( tr ¯ g S ¯ g ) | g + ( n − ( n + 2)8 n ( n − R ¯ g | ˚ S ¯ g | g (cid:21) dv ¯ g ≥ n − (cid:18) Λ ¯ gE + ( n − ( n + 2)8 n ( n − min M R ¯ g (cid:19) Z M | ˚ S ¯ g | g dv ¯ g ≥ due to assumptions on the spectrum of ( − ∆ ¯ gE ) and tr ¯ g S ¯ g = R ¯ g n − is a constant for ≤ n ≤ . Therefore, we conclude that E ¯ g = ( n − S ¯ g = 0 , which shows ¯ g is an Einstein metric. (cid:3) In particular, we have
Corollary 3.1.
Suppose ( M n , ¯ g ) is a closed J -Einstein manifold with non-negative constantscalar curvature and the Einstein operator ∆ ¯ gE on S ( M ) is negative, then ¯ g is a strictlystable Einstein metric.Proof. It is straightforward that ¯ g is an Einstein metric according to Theorem 1.5. Moreover,we obtain inf h ∈ S TT , ¯ g ( M ) \{ } R M h h, − ∆ ¯ gE h i ¯ g dv ¯ g R M | h | g dv ¯ g ≥ inf h ∈ S ( M ) \{ } R M h h, − ∆ ¯ gE h i ¯ g dv ¯ g R M | h | g dv ¯ g > , since S TT , ¯ g $ S ( M ) . Thus the metric ¯ g is strictly stable Einstein by definition. (cid:3) .2. Variations of J -tensor at Einstein metrics. As a geometric symmetric -tensor, J -tensor has a natural connection with Q -curvatureas we have discussed in the introduction. It is crucial to investigate variational properties of J -tensor, when considering variational problems associated to Q -curvature. In this section,we will obtain the first variation of the traceless part of J -tensor at an Einstein metric alongthe T T -direction, which is critical in our further discussion.For simplicity, we may use ′ to denote the first variation in the space of metrics. Thedirection of the variation will be clear from the context. Lemma 3.2.
Suppose ¯ g is an Einstein metric, then ( DE ¯ g ) · ˚ h = −
12 ∆ ¯ gE ˚ h and ( DR ¯ g ) · ˚ h = 0 for any ˚ h ∈ S TT , ¯ g ( M ) .Proof. This is straightforward from first variations of Ricci and scalar curvature in Section2.2 together with facts that E ¯ g = 0 and ˚ h ∈ S TT , ¯ g ( M ) . (cid:3) For Einstein metrics, the connection Laplacian is commutative with first variation:
Lemma 3.3.
Suppose ¯ g is an Einstein metric, then (∆ ¯ g E ) ′ =∆ ¯ g E ′ . Proof.
It is straightforward that ∆ ¯ g E jk =¯ g il ∇ i ∇ l E jk =¯ g il (cid:2) ∂ i ( ∇ l E jk ) − Γ pil ∇ p E jk − Γ pij ∇ l E pk − Γ pik ∇ l E jp (cid:3) =¯ g il (cid:2) ∂ i ( ∂ l E jk − Γ plj E pk − Γ plk E jp ) − Γ pil ∇ p E jk − Γ pij ∇ l E pk − Γ pik ∇ l E jp (cid:3) =¯ g il (cid:2) ∂ i ∂ l E jk − ( ∂ i Γ plj ) E pk − ( ∂ i Γ plk ) E jp − Γ plj ∂ i E pk − Γ plk ∂ i E jp − Γ pil ∇ p E jk − Γ pij ∇ l E pk − Γ pik ∇ l E jp (cid:3) . This shows (∆ ¯ g E jk ) ′ =¯ g il (cid:2) ∂ i ∂ l E ′ jk − ( ∂ i Γ plj ) E ′ pk − ( ∂ i Γ plk ) E ′ jp − Γ plj ∂ i E ′ pk − Γ plk ∂ i E ′ jp − Γ pil ∇ p E ′ jk − Γ pij ∇ l E ′ pk − Γ pik ∇ l E ′ jp (cid:3) =∆ ¯ g E ′ jk , when ¯ g is Einstein. (cid:3) Now the first variation of Bach tensor at an Einstein metric ¯ g is given as follows: Proposition 3.4.
Suppose ¯ g is an Einstein metric, then for any ˚ h ∈ S TT , ¯ g ( M ) , ( DB ¯ g ) · ˚ h = − n − (cid:18) − ∆ ¯ gE + n − n ( n − R ¯ g (cid:19) ( − ∆ ¯ gE ˚ h ) . (3.1) roof. Rewriting the Bach tensor in terms of scalar curvature and traceless Ricci tensor, wehave B g = 1 n − gE E g − n − (cid:18) ∇ g R g − n g ∆ g R g (cid:19) − n − n − E g − n − | E g | g g − R g n ( n − E g . From Lemma 3.3 and E ¯ g = 0 , B ′ ¯ g = 1 n − ¯ gE E ′ ¯ g − n − (cid:18) ∇ g R ′ ¯ g − n ¯ g ∆ ¯ g R ′ ¯ g (cid:19) − R ¯ g n ( n − E ′ ¯ g , where all variations are taken along the T T -direction ˚ h ∈ S TT , ¯ g ( M ) . According to Lemma3.2, we have R ′ ¯ g = 0 and hence ( DB ¯ g ) · ˚ h = − n − (cid:18) − ∆ ¯ gE + n − n ( n − R ¯ g (cid:19) ( − ∆ ¯ gE ˚ h ) . (cid:3) Remark . Note that when n = 4 and Ric ¯ g = 3 λ ¯ g, then equation (3.1) becomes ( DB ¯ g ) · ˚ h = − (cid:0) − ∆ ¯ gE + 2 λ (cid:1) ( − ∆ ¯ gE ˚ h ) . Recall that Bach tensor appears to be the obstruction tensor in the study of ambient space([8]). In fact, Matsumoto has shown that for m th -order obstruction tensor O (2 m )¯ g , its firstvariation at a m -dimensional Einstein metric with Ricci curvature Ric ¯ g = (2 m − λ ¯ g is given by (cid:16) D O (2 m )¯ g (cid:17) · ˚ h = ( − m − m − " m − Y k =0 (cid:0) − ∆ ¯ gE + 2 k (2 m − k − λ (cid:1) ˚ h, for any ˚ h ∈ S TT , ¯ g ( M ) . Please see [15] for further information.For the rest part of ˚ J ¯ g , we have Lemma 3.6.
Suppose ¯ g is an Einstein metric, then ( DT ¯ g ) · ˚ h = n + 2 n − n ( n − R ¯ g ∆ ¯ gE ˚ h for any ˚ h ∈ S TT , ¯ g ( M ) .Proof. Rewriting the tensor T g in terms of scalar curvature R g and traceless Ricci curvature E g , we get T g = n − n − (cid:18) ∇ g R g − n g ∆ g R g (cid:19) + 4( n − n − (cid:18) E g − n | E g | g g (cid:19) − n + 2 n − n ( n − R g E g . Then the result follows from the fact that E ¯ g = 0 and Lemma 3.2. (cid:3) ow combining first variations of B g and T g , we derive Proposition 3.7.
Suppose ¯ g is an Einstein metric, then ( D ˚ J ¯ g ) · ˚ h = 12( n − (cid:18) − ∆ ¯ gE + ( n − ( n + 2)8 n ( n − R ¯ g (cid:19) ( − ∆ ¯ gE ˚ h ) , for any ˚ h ∈ S TT , ¯ g ( M ) .According to Definition 1.6, immediately we obtain the following characterization of theoperator D ˚ J ¯ g for strictly stable Einstein metrics ¯ g : Corollary 3.8.
Suppose ¯ g is an Einstein metric, then D ˚ J ¯ g is a formally self-adjoint operatoron S TT , ¯ g ( M ) . Moreover, D ˚ J ¯ g is a positive operator, if ¯ g is a strictly stable Einstein metricwith non-negative scalar curvature. The key functional and its variations
For completeness, we start with generic J -Einstein metrics instead of more restricted Ein-stein metrics.Suppose ( M n , ¯ g ) is a closed J -Einstein manifold and consider the functional F M, ¯ g ( g ) = V M ( g ) n Z M Q ( g ) dv ¯ g . Note that the volume form dv ¯ g is independent of g and thus the functional F M, ¯ g is scalinginvariant : F M, ¯ g ( c g ) = F M, ¯ g ( g ) for any real number c = 0 .The functional F M, ¯ g is particularly designed for our purpose. This can be glimpsed fromits variational properties. Proposition 4.1.
The J -Einstein metric ¯ g is a critical point of the functional F M, ¯ g . Proof.
For any h ∈ S ( M ) , ( D F M, ¯ g ) · h = V M (¯ g ) n Z M (( DQ ¯ g ) · h ) dv ¯ g + 4 n V M (¯ g ) n − V ′ M, ¯ g Z M Q ¯ g dv ¯ g = V M (¯ g ) n (cid:20)Z M h Γ ¯ g h, i ¯ g dv ¯ g + 2 n (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) V M (¯ g ) − Z M Q ¯ g dv ¯ g (cid:21) = V M (¯ g ) n (cid:20)Z M h h, Γ ∗ ¯ g i ¯ g dv ¯ g + 2 n Q ¯ g Z M ( tr ¯ g h ) dv ¯ g (cid:21) = − M (¯ g ) n Z M (cid:28) h, J ¯ g − n Q ¯ g ¯ g (cid:29) ¯ g dv ¯ g here Q ¯ g := V M (¯ g ) − Z M Q ¯ g dv ¯ g is the average of Q ¯ g on M .According to Definition 1.4, a J -Einstein metric has constant Q -curvature and hence J ¯ g − n Q ¯ g ¯ g = 0 , which implies ¯ g is a critical point of F M, ¯ g . (cid:3) In principle, one can obtain the second variation of F M, ¯ g from a formal calculation. How-ever, due to its complicated expression, the calculation would be rather messy. Instead, withan elegant trick, we can minimize the work through the first variation of J -tensor. Analo-gous to the previous section, we adopt the convention that ′ stands for first and ′′ for secondvariations with certain h ∈ S ( M ) .We start with a useful observation: Lemma 4.2.
For any metric g and h ∈ S ( M ) , tr g ˚ J ′ g = h ˚ J g , ˚ h i g , where ˚ J ′ g is the first variation of traceless J -tensor and ˚ h is the traceless part of h .Proof. Since tr g ˚ J g = 0 , differentiating both sides of the equation gives tr g ˚ J ′ g + ( g − ) ′ · ˚ J g = tr g ˚ J ′ g − h · ˚ J g . That is, tr g ˚ J ′ g = h ˚ J g , h i g = (cid:28) ˚ J g , ˚ h + 1 n ( tr g h ) g (cid:29) g = h ˚ J g , ˚ h i g . (cid:3) With the above observation, we can express the integral of second variation of Q -curvaturein a compact form. Proposition 4.3.
Suppose ¯ g is a J -Einstein metric, then we have Z M Q ′′ ¯ g dv ¯ g = − Z M (cid:20) h ˚ J ′ ¯ g , ˚ h i ¯ g − n Q ¯ g | ˚ h | g + (cid:18) n + 44 n Q ′ ¯ g + n − n Q ¯ g ( tr ¯ g h ) (cid:19) ( tr ¯ g h ) (cid:21) dv ¯ g holds for any h ∈ S ( M ) . roof. We start with an arbitrary Riemannian metric g on M . It is straightforward that forany h ∈ S ( M ) , (cid:18)Z M Q g dv g (cid:19) ′′ = (cid:18)Z M Q ′ g dv g + Z M Q g ( dv g ) ′ (cid:19) ′ = (cid:18)Z M h Γ ∗ g , h i g dv g (cid:19) ′ + Z M Q ′ g ( dv g ) ′ + Z M Q g ( dv g ) ′′ = − Z M (cid:20) h J ′ g , h i g + 2 D(cid:0) g − (cid:1) ′ , J g × h E g + 12 h J g , h i g ( tr g h ) (cid:21) dv g + Z M Q ′ g ( dv g ) ′ + Z M Q g ( dv g ) ′′ = − Z M (cid:20) h J ′ g , h i g − (cid:10) J g , h (cid:11) g + 12 h J g , h i g ( tr g h ) (cid:21) dv g + Z M Q ′ g ( dv g ) ′ + Z M Q g ( dv g ) ′′ . Furthermore, the decomposition J g = ˚ J g + 1 n Q g g yields J ′ g = ˚ J ′ g + 1 n Q ′ g g + 1 n Q g h. Together with variational formulas in Section 2.2, we have Z M Q ′′ g dv g = (cid:18)Z M Q g dv g (cid:19) ′′ − Z M Q ′ g ( dv g ) ′ − Z M Q g ( dv g ) ′′ = − Z M "(cid:28) ˚ J ′ g + 1 n Q ′ g g + 1 n Q g h, h (cid:29) g − (cid:28) ˚ J g + 1 n Q g g, h (cid:29) g dv g − Z M "(cid:28) ˚ J g + 1 n Q g g, h (cid:29) g + 12 Q ′ g ( tr g h ) dv g = − Z M " h ˚ J ′ g , h i g − (cid:28) ˚ J g + 12 n Q g g, h (cid:29) g + 12 (cid:18) n + 42 n Q ′ g + 1 n Q g ( tr g h ) + h ˚ J g , h i g (cid:19) ( tr g h ) dv g . rom the decomposition h = ˚ h + n ( tr g h ) g and Lemma 4.2, we can further simplify theexpression to the following form: Z M Q ′′ g dv g = − Z M "(cid:28) ˚ J ′ g , ˚ h + 1 n ( tr g h ) g (cid:29) g − (cid:28) ˚ J g + 12 n Q g g, ˚ h + 2 n ( tr g h )˚ h + 1 n ( tr g h ) g (cid:29) g dv g − Z M " n + 42 n Q ′ g + 1 n Q g ( tr g h ) + (cid:28) ˚ J g , ˚ h + 1 n ( tr g h ) g (cid:29) g ( tr g h ) dv g = − Z M " h ˚ J ′ g , ˚ h i g − (cid:28) ˚ J g + 12 n Q g g, ˚ h (cid:29) g dv g − Z M (cid:20) n + 42 n Q ′ g + ( n − n Q g ( tr g h ) + ( n − n h ˚ J g , ˚ h i g (cid:21) ( tr g h ) dv g . In particular for J -Einstein ¯ g , we have ˚ J ¯ g = 0 and hence Z M Q ′′ ¯ g dv ¯ g = − Z M (cid:20) h ˚ J ′ ¯ g , ˚ h i ¯ g − n Q ¯ g | ˚ h | g + (cid:18) n + 44 n Q ′ ¯ g + n − n Q ¯ g ( tr ¯ g h ) (cid:19) ( tr ¯ g h ) (cid:21) dv ¯ g . (cid:3) Now we calculate the second variation of the functional F M, ¯ g at a J -Einstein metric ¯ g . Proposition 4.4.
For n ≥ , suppose ( M n , ¯ g ) is J -Einstein, then for any h ∈ S ( M ) wehave V M (¯ g ) − n (cid:0) ( D F M, ¯ g ) · ( h, h ) (cid:1) = − Z M (cid:20) h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g + 1 n (cid:18) tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h ) + n + 44 (Γ ¯ g ˚ h ) (cid:19) ( tr ¯ g h ) (cid:21) dv ¯ g − n + 42 n Z M (cid:2) ( tr ¯ g h − tr ¯ g h ) L ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:3) dv ¯ g , where the operator L ¯ g is defined to be L ¯ g u := tr ¯ g Γ ∗ ¯ g u = 12 (cid:18) P ¯ g − n + 42 Q ¯ g (cid:19) u. roof. Applying variational formulae of volume in Section 2.2, we have ( D F M, ¯ g ) · ( h, h )= (cid:18) V M ( g ) n Z M Q g dv ¯ g (cid:19) ′′ = (cid:18) (V M ( g ) n ) ′ Z M Q g dv ¯ g + V M ( g ) n Z M Q ′ g dv ¯ g (cid:19) ′ = V M (¯ g ) n Z M Q ′′ ¯ g dv ¯ g + 8 n V M (¯ g ) n − V ′ M, ¯ g Z M Q ′ ¯ g dv ¯ g + 4 n V M (¯ g ) n − V ′′ M, ¯ g Z M Q ¯ g dv ¯ g − n − n V M (¯ g ) n − (V ′ M, ¯ g ) Z M Q ¯ g dv ¯ g = V M (¯ g ) n Z M Q ′′ ¯ g dv ¯ g − n V M (¯ g ) − nn Z M ( tr ¯ g h ) dv ¯ g Z M h J ¯ g , h i ¯ g dv ¯ g + 1 n V M (¯ g ) n Q ¯ g "Z M (cid:18) n − n ( tr ¯ g h ) − | ˚ h | g (cid:19) dv ¯ g − n − n V M (¯ g ) − (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) , where we used the fact Q ¯ g is a constant, when ¯ g is J -Einstein.Now applying Lemma 4.3, it can be simplified as V M (¯ g ) − n (cid:0) ( D F M, ¯ g ) · ( h, h ) (cid:1) = − Z M h ˚ J ′ ¯ g , ˚ h i ¯ g dv ¯ g − n + 42 n Z M Q ′ ¯ g ( tr ¯ g h ) dv ¯ g − n + 4 n Q ¯ g V M (¯ g ) − (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) . Rewriting its first term as − Z M h ˚ J ′ ¯ g , ˚ h i ¯ g dv ¯ g = − Z M (cid:28) ( D ˚ J ¯ g ) · (cid:18) ˚ h + 1 n ( tr ¯ g h )¯ g (cid:19) , ˚ h (cid:29) ¯ g dv ¯ g = − Z M h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g dv ¯ g − n Z M [ tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h )]( tr ¯ g h ) dv ¯ g , we obtain V M (¯ g ) − n (cid:0) ( D F M, ¯ g ) · ( h, h ) (cid:1) = − Z M h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g dv ¯ g − n Z M [ tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h )]( tr ¯ g h ) dv ¯ g − n + 42 n Z M (cid:2) Q ′ ¯ g ( tr ¯ g h ) (cid:3) dv ¯ g − n + 4 n Q ¯ g V M (¯ g ) − (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) = − Z M h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g dv ¯ g − n Z M [ tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h )]( tr ¯ g h ) dv ¯ g − n + 42 n Z M h ( tr ¯ g h )(Γ ¯ g ˚ h ) i dv ¯ g − n + 42 n Z M ( tr ¯ g h )[ tr ¯ g Γ ∗ ¯ g ( tr ¯ g h )] dv ¯ g − n + 4 n Q ¯ g V M (¯ g ) − (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) . Denote ˚ P ¯ g := P ¯ g − n − Q ¯ g o be the divergence part of Paneitz operator. Clearly the space of constants are containedin ker ˚ P ¯ g . This implies − n + 42 n Z M ( tr ¯ g h )[ tr ¯ g Γ ∗ ¯ g ( tr ¯ g h )] dv ¯ g − n + 4 n Q ¯ g V M (¯ g ) − (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) = − n + 44 n Z M ( tr ¯ g h ) h(cid:16) ˚ P ¯ g − Q ¯ g (cid:17) ( tr ¯ g h ) i dv ¯ g − n + 4 n Q ¯ g V M (¯ g ) − (cid:18)Z M ( tr ¯ g h ) dv ¯ g (cid:19) = − n + 44 n Z M ( tr ¯ g h ) (cid:16) ˚ P ¯ g ( tr ¯ g h ) (cid:17) dv ¯ g + n + 4 n Q ¯ g Z M ( tr ¯ g h − tr ¯ g h ) dv ¯ g = − n + 44 n Z M h ( tr ¯ g h − tr ¯ g h ) (cid:16) ˚ P ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:17) − Q ¯ g ( tr ¯ g h − tr ¯ g h ) i dv ¯ g = − n + 42 n Z M (cid:2) ( tr ¯ g h − tr ¯ g h ) L ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:3) dv ¯ g . Therefore, V M (¯ g ) − n (cid:0) ( D F M, ¯ g ) · ( h, h ) (cid:1) = − Z M (cid:20) h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g + 1 n [ tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h )]( tr ¯ g h ) + n + 44 n ( tr ¯ g h )(Γ ¯ g ˚ h ) (cid:21) dv ¯ g − n + 42 n Z M (cid:2) ( tr ¯ g h − tr ¯ g h ) L ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:3) dv ¯ g . = − Z M (cid:20) h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g + 1 n (cid:18) tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h ) + n + 44 (Γ ¯ g ˚ h ) (cid:19) ( tr ¯ g h ) (cid:21) dv ¯ g − n + 42 n Z M (cid:2) ( tr ¯ g h − tr ¯ g h ) L ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:3) dv ¯ g . (cid:3) Remark . In general, for a J -Einstein metric ¯ g , the term Z M [ tr ¯ g (( D ˚ J ¯ g ) ∗ · ˚ h )]( tr ¯ g h ) dv ¯ g in the above proposition may not necessarily vanish unless the dimension n = 4 . But if ¯ g isan Einstein metric, this term would be zero due to the operator D ˚ J ¯ g is formally self-adjointand Lemma 4.2. This case is addressed in Corollary 4.7.In order to remove the crossing term in the previous proposition and obtain an elegantvariational formula for Einstein metrics, we first rewrite the operator Γ g in a nice way. Proposition 4.6.
Suppose g is an arbitrary Riemannian metric and h = ˚ h + 1 n ( tr g h ) g ∈ S TT ,g ( M ) ⊕ ( C ∞ ( M ) · g ) , then Γ g h = div g [ U g (˚ h )] − J g · ˚ h + 1 n L g ( tr g h ) , here U g (˚ h ) := 12( n − n − h n ∇ g (˚ S g · ˚ h ) − n − h ij ∇ g ˚ S ijg + ( n + 4 n − h ( ∇ g ( tr g S g )) i . Proof.
By the relation between Ricci and Schouten tensor
Ric g = ( n − S g + ( tr g S g ) g, we can express Γ g ˚ h in terms of Schouten tensor Γ g ˚ h = n − n −
1) ∆ g (˚ S g · ˚ h ) + ∇ ( tr g S g ) · ˚ h + 2 n − S g · ∆ g ˚ h + 4 n − S g · ( Rm g · ˚ h ) − ( n − ( n + 2)2 n ( n −
1) ( tr g S g )(˚ S g · ˚ h ) , which has simpler coefficients instead. Recall the traceless part of J -tensor is given by ˚ J g = − n − (cid:18) B g + n − n − T g (cid:19) = − n − (cid:18) ∆ g ˚ S g + n − n + 124( n − (cid:18) ∇ g ( tr g S g ) − n g ∆ g ( tr g S g ) (cid:19) + 2 Rm g · ˚ S g (cid:19) + ( n − ( n + 2)4 n ( n −
1) ( tr g S g )˚ S g + 2 n | ˚ S g | g g. Now we have Γ g ˚ h + 2 ˚ J g · ˚ h = n − n −
1) ∆ g (˚ S g · ˚ h ) + n + 4 n − n − n − ∇ ( tr g S g ) · ˚ h + 2 n − h ˚ S g · ∆ g ˚ h − (∆ g ˚ S g ) · ˚ h i = n − n − div g (cid:20) ∇ g (˚ S g · ˚ h ) + n + 4 n − n − ˚ h ( ∇ g ( tr g S g )) + 4( n − n − (cid:16) ˚ S ijg ∇ g ˚ h ij − ˚ h ij ∇ g ˚ S ijg (cid:17)(cid:21) = 12( n − n − div g h n ∇ g (˚ S g · ˚ h ) − n − h ij ∇ g ˚ S ijg + ( n + 4 n − h ( ∇ g ( tr g S g )) i = div g [ U g (˚ h )] , where U g (˚ h ) = 12( n − n − h n ∇ g (˚ S g · ˚ h ) − n − h ij ∇ g ˚ S ijg + ( n + 4 n − h ( ∇ g ( tr g S g )) i That is, Γ g ˚ h = div g [ U g (˚ h )] − J g · ˚ h. As for the trace part, we have Z M (cid:20) u Γ g (cid:18) n ( tr g h ) g (cid:19)(cid:21) dv g = Z M (cid:28) Γ ∗ g u, n ( tr g h ) g (cid:29) g dv g = 1 n Z M (cid:2) ( tr g h )( tr g Γ ∗ g u ) (cid:3) dv g for any u ∈ C ∞ ( M ) . Recall that L g = tr g Γ ∗ g s defined in Proposition 4.4 and L g is formally self-adjoint, then Z M (cid:20) u Γ g (cid:18) n ( tr g h ) g (cid:19)(cid:21) dv g = 1 n Z M [( tr g h )( L g u )] dv g = 1 n Z M [ u L g ( tr g h )] dv g . Since u is arbitrary, we conclude that Γ g (cid:18) n ( tr g h ) g (cid:19) = 1 n L g ( tr g h ) . Combining these two parts, we obtain Γ g h = div g [ U g (˚ h )] − J g · ˚ h + 1 n L g ( tr g h ) . (cid:3) In particular, when ¯ g is restricted to be an Einstein metric, the second variation of F M, ¯ g can be expressed in an elegant way: Corollary 4.7.
Suppose ( M n , ¯ g ) is an Einstein manifold, then ( D F M, ¯ g ) · ( h, h )= − M (¯ g ) n (cid:20)Z M h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g dv ¯ g + n + 44 n Z M (cid:2) ( tr ¯ g h − tr ¯ g h ) L ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:3) dv ¯ g (cid:21) , for any h = ˚ h + n ( tr ¯ g h )¯ g ∈ S TT , ¯ g ( M ) ⊕ ( C ∞ ( M ) · ¯ g ) .Proof. This result follows from Corollary 3.8, Lemma 4.2, and the fact that U ¯ g (˚ h ) = 0 and ˚ J ¯ g = 0 when ¯ g is an Einstein metric. (cid:3) Volume comparison with respect to Q -curvature In this section, we give the proof of our main results. As the first step, we recall somefundamental results involved (c.f. [1, 18]):
Lemma 5.1 (Lichnerowicz-Obata’s eigenvalue estimate) . Suppose ( M n , ¯ g ) is an n -dimensionalclosed Riemannian manifold with Ric ¯ g ≥ ( n − λ ¯ g, where λ > is a constant. Then for any function u ∈ C ∞ ( M ) \{ } with Z M udv ¯ g = 0 , we have Z M | du | dv ¯ g ≥ nλ Z M u dv ¯ g , where equality holds if and only if ( M n , ¯ g ) is isometric to the round sphere S n ( r ) with radius r = √ λ . emma 5.2 (Berger-Ebin’s splitting lemma for Einstein manifolds) . Suppose ( M n , ¯ g ) is an n -dimensional closed Einstein manifold with Ricci curvature Ric ¯ g = ( n − λ ¯ g, then we have the direct sum decomposition S ( M ) = Im L ¯ g ⊕ ( C ∞ ( M ) · ¯ g ) ⊕ S TT , ¯ g ( M ) unless ( M n , ¯ g ) is isometric to the round sphere S n ( r ) up to a scaling, where L ¯ g is the Liederivative. For the round sphere S n ( r ) with radius r = √ λ , we have S ( M ) = Im L ¯ g ⊕ ( E ⊥ nλ · ¯ g ) ⊕ S TT , ¯ g ( M ) , where E nλ := { u ∈ C ∞ ( S n ( r )) | ∆ S n ( r ) u + nλu = 0 } is the space of first eigenfunctions for the spherical metric and E ⊥ nλ is its L -orthogonalcomplement. With the aid of
Implicit Function Theorem , Berger-Ebin’s splitting lemma suggests thatone can define a concept named local slice S ¯ g , which is very helpful in understanding thelocal structure of Einstein metrics in M . Simply speaking, a local slice is a set of equivalentclasses of metrics near the reference metric ¯ g modulo diffeomorphisms. The process of pullingback metrics on the local slice is also known as gauge fixing .The above splitting lemma does not provide an orthogonal decomposition despite being adirect sum decomposition. To overcome this issue, we need a refined decomposition whichinvolves the splitting of vector fields as well. As a result, we obtain the following improvedversion of the traditional Ebin-Palais slice theorem . The proof is very similar to the tradi-tional one (see [3, 18]). It seems like such a treatment did not appear in the literature to thebest of our knowledge and we hope it would benefit researches of similar topics.
Theorem 5.3 (Ebin-Palais slice theorem) . Suppose ( M n , ¯ g ) is a closed n -dimensional Ein-stein manifold with Ricci curvature tensor Ric ¯ g = ( n − λ ¯ g, where λ ∈ R is a constant. There exists a local slice S ¯ g though ¯ g in M . That is, for a fixedreal number p > n , one can find a constant ε > such that for any metric g ∈ M with || g − ¯ g || W ,p ( M, ¯ g ) < ε , there is a diffeomorphism ϕ ∈ D ( M ) with ϕ ∗ g ∈ S ¯ g . Moreover, for asmooth local slice S ¯ g , we have the decomposition S ( M ) = T ¯ g S ¯ g ⊕ ( T ¯ g S ¯ g ) ⊥ , where the tangent space of S ¯ g at ¯ g and its L -orthogonal complement are given by T ¯ g S ¯ g = S TT , ¯ g ( M ) ⊕ ( C ∞ ( M ) · ¯ g ) and ( T ¯ g S ¯ g ) ⊥ = (cid:8) L ¯ g ( X ) | h X, ∇ ¯ g u i L ( M, ¯ g ) = 0 , ∀ u ∈ C ∞ ( M ) (cid:9) , when ( M n , ¯ g ) is not isometric to the round sphere S n ( r ) up to a scaling; T ¯ g S ¯ g = S TT , ¯ g ( M ) ⊕ ( E ⊥ nλ · ¯ g ) nd ( T ¯ g S ¯ g ) ⊥ = (cid:8) L ¯ g ( X ) | h X, ∇ ¯ g u i L ( M, ¯ g ) = 0 , ∀ u ∈ E ⊥ nλ (cid:9) , when ( M n , ¯ g ) is isometric to the round sphere S n ( r ) with r = √ λ . Here E nλ = { u ∈ C ∞ ( S n ( r )) : ∆ S n ( r ) u + nλu = 0 } is the space of first eigenfunctions for the spherical metric. In order to estimate the second variation of F M, ¯ g , we also need to investigate the analyticproperties of the operator L ¯ g (as defined in Proposition 4.4): Proposition 5.4.
Suppose ¯ g is an Einstein metric with Ricci curvature Ric ¯ g = ( n − λ ¯ g, where λ ≥ is a constant, then the operator L ¯ g is a non-negative operator. Moreover, L ¯ g admits non-trivial kernel when • λ > and ¯ g is spherical: ker L ¯ g = E nλ , • λ = 0 and ¯ g is Ricci-flat: ker L ¯ g = R . Proof.
By definition, L ¯ g u = 12 (cid:18) P ¯ g − n + 42 Q ¯ g (cid:19) u = 12 (cid:18) − ∆ ¯ g + ( n − n + 2)2 λ (cid:19) ( − ∆ ¯ g − nλ ) u. For λ > , the first eigenvalue of ( − ∆ ¯ g ) is at least nλ by Lemma 5.1, which implies theoperator L ¯ g is non-negative and ker L ¯ g = ker ( − ∆ ¯ g − nλ ) . This shows L ¯ g has non-trivial kernel if and only if ¯ g is spherical and ker L ¯ g consisted of firsteigenfunctions of ∆ ¯ g .For λ = 0 , L ¯ g u = 12 ∆ g u. It is clear that L ¯ g is non-negative and ker L ¯ g = ker ∆ ¯ g = R . (cid:3) With these preparations, we summarize variational properties of F M, ¯ g at a strictly stableEinstein metric ¯ g : roposition 5.5. Suppose ( M n , ¯ g ) is a strictly stable Einstein manifold with Ric ¯ g = ( n − λ ¯ g, where λ ≥ is a constant, then ¯ g is a critical point of F M, ¯ g and ( D F M, ¯ g ) · ( h, h ) ≤ for any h = ˚ h + n ( tr ¯ g h )¯ g ∈ S TT , ¯ g ( M ) ⊕ ( C ∞ ( M ) · ¯ g ) . Moreover, the equality holds if and onlyif • h ∈ R ¯ g , when ( M n , ¯ g ) is not isometric to the round sphere up to a rescaling of themetric. • h ∈ ( R ⊕ E nλ )¯ g , when ( M n , ¯ g ) is isometric to the round sphere S n ( r ) with radius r = √ λ ,where E nλ := { u ∈ C ∞ ( S n ( r )) | ∆ S n ( r ) u + nλu = 0 } is the space of first eigenfunctions for the spherical metric. Proof.
According to Proposition 4.1, we can conclude that ¯ g is a critical point of F M, ¯ g , sinceEinstein metrics are J -Einstein. Recall in Corollary 4.7, we showed that ( D F M, ¯ g ) · ( h, h )= − M (¯ g ) n (cid:20)Z M h ˚ h, ( D ˚ J ¯ g ) · ˚ h i ¯ g dv ¯ g + n + 44 n Z M (cid:2) ( tr ¯ g h − tr ¯ g h ) L ¯ g ( tr ¯ g h − tr ¯ g h ) (cid:3) dv ¯ g (cid:21) holds for any h ∈ S TT , ¯ g ( M ) ⊕ ( C ∞ ( M ) · ¯ g ) . It is obvious that D F M, ¯ g is non-positive definiteaccording to Corollary 3.8 and Proposition 5.4. Furthermore, D F M, ¯ g vanishes if and only ˚ h = 0 and ( tr ¯ g h − tr ¯ g h ) ∈ ker L ¯ g . Now the conclusion follows from Proposition 5.4. (cid:3)
Another fundamental result we need is the following version of
Morse lemma on Banachmanifold for degenerate functions:
Lemma 5.6 (Fisher-Marsden [7]) . Let P be a Banach manifold and f : P → R a C func-tion. Suppose that Q ⊂ P is a submanifold, f = 0 and df = 0 on Q and that there is asmooth normal bundle neighborhood of Q such that if E x is the normal complement to T x Q in T x P then d f ( x ) is weakly negative definite on E x ( i.e. d f ( x )( v, v ) ≤ with equalityonly if v = 0) . Let hh , ii x be a weak Riemannian structure with a smooth connection andassume that f has a smooth hh , ii x -gradient, Y ( x ) . Assume DY ( x ) maps E x to E x and isan isomorphism for x ∈ Q . Then there is a neighborhood U of Q such that y ∈ U , f ( y ) ≥ implies y ∈ Q . According to Theorem 5.3, we can find a local slice S ¯ g through ¯ g and identify Q ¯ g to bethe submanifold of S ¯ g consisted of homothetic metrics, that is, Q ¯ g := { c ¯ g ∈ S ¯ g | c = 0 } . onsider the restriction of F M, ¯ g on the local slice S ¯ g , denoted by F M, ¯ g | S ¯ g . Applying theprevious Morse lemma, we obtain the following rigidity result: Proposition 5.7.
Suppose ( M n , ¯ g ) is a strictly stable Einstein manifold with Ricci curvature Ric ¯ g = ( n − λ ¯ g, where λ ≥ is a constant. There is a neighborhood of ¯ g in the local slice S ¯ g , denoted by U ¯ g ,such that any metric g s ∈ U ¯ g satisfying F M, ¯ g | S ¯ g ( g s ) ≥ F M, ¯ g | S ¯ g (¯ g ) implies that g s = c ¯ g for some constant c > . Proof.
From Proposition 5.5, we conclude that ¯ g is a critical point of F M, ¯ g | S ¯ g and D F M, ¯ g | S ¯ g is non-positive definite on T ¯ g S ¯ g . Moreover, it is obvious that D F M, ¯ g | S ¯ g is degenerate if andonly if when restricted on T ¯ g Q ¯ g = R ¯ g. Let E ¯ g be the L -orthogonal complement of T ¯ g Q ¯ g in T ¯ g S ¯ g . By Theorem 5.3, we can identify E ¯ g = (cid:26) h ∈ S TT , ¯ g ( M ) ⊕ ( C ∞ ( M ) · ¯ g ) (cid:12)(cid:12)(cid:12)(cid:12) Z M ( tr ¯ g h ) dv ¯ g = 0 (cid:27) , if ¯ g is not spherical; E ¯ g = (cid:26) h ∈ S TT , ¯ g ( M ) ⊕ ( E ⊥ nλ · ¯ g ) (cid:12)(cid:12)(cid:12)(cid:12) Z M ( tr ¯ g h ) dv ¯ g = 0 (cid:27) , if ¯ g is spherical. Therefore, D F M, ¯ g | S ¯ g is strictly negative definite on E ¯ g .We introduce a weak Riemannian structure hh h, h ii g s := Z M [ h h, h i g s + h∇ g s h, ∇ g s h i g s ] dv g s = Z M h (1 − ∆ g s ) h, h i g s dv g s on S ¯ g . As in [20], it has a smooth connection and the hh , ii g s -gradient of F M, ¯ g (cid:12)(cid:12) S ¯ g is givenby Y ( g s ) = P g s (1 − ∆ g s ) − (cid:20) V M ( g s ) n (cid:18) Γ ∗ g s ( ρ g s ) + 2 n g s V M ( g s ) − n +4 n F M, ¯ g ( g s ) (cid:19)(cid:21) , where P g s is the orthogonal projection to T g s S ¯ g and ρ g s > is a smooth function on M satisfying dv ¯ g = ρ g s dv g s . Obviously, Y ( g s ) is a smooth vector field on S ¯ g . Now we define anauxiliary vector field on S ¯ g , Z ( g s ) := V M ( g s ) n (cid:18) Γ ∗ g s ( ρ g s ) + 2 n g s V M ( g s ) − n +4 n F M, ¯ g ( g s ) (cid:19) . It is straightforward that Z (¯ g ) = 0 due to the fact that ¯ g is Einstein and furthermore, ( DZ ¯ g ) · h = ( D F M, ¯ g | S ¯ g ) · ( h, · ) for any h ∈ E ¯ g . Thus we have DY ¯ g = P ¯ g (1 − ∆ ¯ g ) − ( DZ ¯ g ) , which implies DY ¯ g is an isomorphism on E ¯ g due to the reason that D F M, ¯ g | S ¯ g is strictlynegative definite on E ¯ g from previous discussions. ccording to Lemma 5.6, we can find a neighborhood U ¯ g ⊂ S ¯ g such that any metric g s ∈ U ¯ g satisfying F M, ¯ g | S ¯ g ( g s ) ≥ F M, ¯ g | S ¯ g (¯ g ) = F M, ¯ g (¯ g ) implies that g s ∈ Q ¯ g , which means we can find a constant c > such that g s = c ¯ g . (cid:3) Now we can prove our volume comparison theorem:
Proof of Theorem 1.7.
Applying Theorem 5.3, we can find a positive constant ε < ε suchthat for any metric ˆ g satisfies that || ˆ g − ¯ g || C ( M, ¯ g ) < ε , there exists a diffeomorphism ϕ such that ϕ ∗ ˆ g ∈ U ¯ g ⊆ S ¯ g , where U ¯ g is defined in Proposition5.7.For λ > , suppose g is a Riemannian metric on M satisfying Q g ≥ Q ¯ g and || g − ¯ g || C ( M, ¯ g ) < ε , but with reversed volume comparison : V M ( g ) ≥ V M (¯ g ) . (5.1)We are going to show that g has to be isometric to ¯ g and hence the claimed volume com-parison holds.According to the argument in the previous paragraph, there exists a diffeomorphism ϕ such that ϕ ∗ g ∈ U ¯ g ⊆ S ¯ g and F M, ¯ g | S ¯ g ( ϕ ∗ g ) = V M ( ϕ ∗ g ) n Z M ( Q g ◦ ϕ ) dv ¯ g ≥ V M (¯ g ) n Z M Q (¯ g ) dv ¯ g = F M, ¯ g | S ¯ g (¯ g ) due to our assumptions and the fact that Q ¯ g is a constant. Thus, we conclude that ϕ ∗ g = c ¯ g for some positive c ∈ R by Proposition 5.7. Now the reversed volume comparison (5.1)becomes V M ( g ) = V M ( ϕ ∗ g ) = c n V M (¯ g ) ≥ V M (¯ g ) , which implies c ≥ . However, the curvature comparison assumption implies c ≤ , since Q ¯ g = Q ¯ g ◦ ϕ ≤ Q g ◦ ϕ = Q ϕ ∗ g = c − Q ¯ g . Therefore, ϕ ∗ g = ¯ g and it concludes the theorem. (cid:3) With the same idea, we can prove the rigidity of strictly stable Ricci-flat manifolds:
Proof of Theorem 1.11.
Similar to the proof of Theorem 1.7, we can find an ε > such thatfor any metric ˆ g satisfies || ˆ g − ¯ g || C ( M, ¯ g ) < ε , there is a diffeomorphism ϕ such that ϕ ∗ ˆ g ∈ U ¯ g ⊂ S ¯ g , where U ¯ g is given by Proposition 5.7.Suppose g is a metric satisfying Q g ≥ nd || g − ¯ g || C ( M, ¯ g ) < ε , then we can find a diffeomorphism ϕ such that ϕ ∗ g ∈ U ¯ g and F M, ¯ g | S ¯ g ( ϕ ∗ g ) = V M ( ϕ ∗ g ) n Z M ( Q g ◦ ϕ ) dv ¯ g ≥ . However, the metric ¯ g is Ricci-flat and hence F M, ¯ g | S ¯ g (¯ g ) = V M (¯ g ) n Z M Q (¯ g ) dv ¯ g = 0 . This shows ϕ ∗ g = c ¯ g for some positive constant c ∈ R by Proposition 5.7, which means g is homothetic to ¯ g up to a diffeomorphism. (cid:3) Remarks and further discussions
In this section, we address some important observations and remarks regarding our maintheorem. First, we make a comment on the stability assumption:
Remark . The stability condition in Theorem 1.7 is necessary. This is the same phenom-enon observed in [20] for the volume comparison of scalar curvature.Let ¯ g be the canonical product metric on S × S × S . It is well-known that this manifoldis unstable (c.f [10]). Consider the metric g t = (1 + t ) − g S + (1 − t ) − g S + (1 + t ) − g S with t ∈ (0 , sufficiently small. Then its Q -curvature is given by Q g t = 150 (cid:0) − t + 7 t + 48 (cid:1) > Q ¯ g However, its volume satisfies V M ( g t ) = (1 − t ) − V M (¯ g ) > V M (¯ g ) . It shows that the volume comparison fails in this case. In fact, the volume comparison isnot expected to hold for unstable Einstein manifolds due to Corollary 4.7.Now we turn to the locality assumption. For general dimensions n ≥ , we have proved avolume comparison result for metrics sufficiently closed to a strictly stable positive Einsteinmetric. It turns out that for dimension n = 4 , we do have a global volume comparison resultas follows: Proposition 6.2.
Let ( M , ¯ g ) be a closed -dimensional locally conformally flat Riemannianmanifold with positive constant Q -curvature. Then for any metric g on M satisfies Q g ≥ Q ¯ g ointwisely on M , we have V M ( g ) ≤ V M (¯ g ) − Q ¯ g || W g || L ( M,g ) ≤ V M (¯ g ) , where equality holds if and only if g is also locally conformally flat. Proof.
By Gauss-Bonnet-Chern formula, π χ ( M ) = Z M Q g dv g + 14 Z M | W g | dv g = Z M Q ¯ g dv ¯ g = Q ¯ g V M (¯ g ) . Then V M (¯ g ) = Q − g (cid:18)Z M Q g dv g + 14 Z M | W g | dv g (cid:19) ≥ V M ( g ) + 14 Q ¯ g Z M | W | g dv g . (cid:3) As a straightforward application, we have
Corollary 6.3.
Let ( M , ¯ g ) be either the standard round -sphere or a closed hyperbolic -manifold. Then for any metrics g satisfies Q g ≥ Q ¯ g , we have V M ( g ) ≤ V M (¯ g ) . Moreover, in case of the round -sphere, the equality holds if and and only if g is isometricto the spherical metric ¯ g .Proof. We only need to prove the rigidity part of -sphere. According to Proposition 6.2, themetric g has to be locally conformally flat and hence g ∈ [¯ g ] by Kuiper’s theorem [11], since S is simply connected. On the other hand, due to our assumption Q g ≥ Q ¯ g and Gauss-Bonnet-Chern formula, we have Q g = Q ¯ g . Now the conclusion follows from a uniquenessresult in [5, 12, 19]. (cid:3) Based on this global volume comparison observed above for -dimensional hyperbolicmanifolds, we would like to propose the following conjecture: Conjecture 6.4.
For any n ≥ , let ( M n , ¯ g ) be a closed hyperbolic manifold. Suppose g isa metric on M with Q g ≥ Q ¯ g , then we have V M ( g ) ≤ V M (¯ g ) . Remark . This is a corresponding version of
Schoen’s conjecture on scalar curvature (see[20] for more details). As a first step, we would be interested in the question that whetherthis conjecture holds for metrics C -closed to the hyperbolic metric ¯ g . In this case, it dependson a further research on the spectrum of the operator L ¯ g . eferences [1] A. L. Besse, Einstein manifolds , Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the1987 edition.[2] T. P. Branson,
Differential operators canonically associated to a conformal structure , Math. Scand. (1985), no. 2, 293–345.[3] S. Brendle and F. C. Marques, Scalar curvature rigidity of geodesic balls in S n , J. Diff. Geom. (2011),379–394.[4] J. S. Case, Y.-J. Lin, and W. Yuan, Conformally variational Riemannian invariants , Trans. Amer.Math. Soc. (2019), no. 11, 8217–8254.[5] S.-Y. A. Chang and P. C. Yang,
On uniqueness of solutions of n -th order differential equations inconformal geometry , Math. Res. Lett. (1997), 91–102.[6] J. Corvino, Michael Eichmair, and Pengzi Miao, Deformation of scalar curvature and volume , Math.Ann. (2013), no. 2, 551–584.[7] A. E. Fischer and J. E. Marsden,
Deformations of the scalar curvature , Duke Math. J. (1975), no. 3,519–547.[8] C. Robin Graham and Kengo Hirachi, The ambient obstruction tensor and Q -curvature , AdS/CFTcorrespondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys.,vol. 8, Eur. Math. Soc., Zürich, 2005, pp. 59–71.[9] F.-B. Hang and P. C. Yang, Lectures on the fourth-order Q curvature equation , Geometric analysisaround scalar curvatures, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 31, World Sci.Publ., Hackensack, NJ, 2016, pp. 1–33.[10] K. Kröncke, Stability of Einstein Manifolds , 2014. Thesis (Ph.D.)–Universität Potsdam, URL http://opus.kobv.de/ubp/volltexte/2014/6963/ .[11] N. H. Kuiper,
On conformally flat spaces in the large , Ann. Math. (1949), 916–924.[12] C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in R n , Comment.Math. Helv. (1998), no. 4, 206–231.[13] Y.-J. Lin and W. Yuan, Deformations of Q-curvature I , Calc. Var. Partial Differential Equations (2016), no. 4, Art. 101, 29.[14] , A symmetric 2-tensor canonically associated to Q -curvature and its applications , Pacific J.Math. (2017), no. 2, 425–438.[15] Y. Matsumoto, A GJMS construction for 2-tensors and the second variation of the total Q -curvature ,Pacific J. Math. (2013), no. 2, 437–455.[16] M. Obata, The conjectures on conformal transformations of Riemannian manifolds , J. Diff. Geom. (1971/72), 247–258.[17] S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannianmanifolds (summary) , SIGMA Symmetry Integrability Geom. Methods Appl. (2008), Paper 036, 3.[18] J. A. Viaclovsky, Critical metrics for Riemannian curvature functionals , IAS/Park City MathematicsSeries (2016), 195–274.[19] X. Xu,
Classification of solutions of certain fourth order nonlinear elliptic equations in R , Pacific J. ofMath. (2006), no. 2, 361–378.[20] W. Yuan, Volume comparison with respect to scalar curvature , arXiv:1609.08849, submitted (2021). (Yueh-Ju Lin) Department of Mathematics, Statistics, and Physics, Wichita State Univer-sity, 1845 Fairmount Street, Wichita, KS 67260, USA
Email address : [email protected] (Wei Yuan) Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong510275, China Email address : [email protected]@mail.sysu.edu.cn