The Second Variational Formula For the Functional $\int v^{(6)}(g)dV_g$

In this note, we compute the second variational formula for the functional $\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict local maximum within its conformal class, unless the manifold is isometric to round sphere with the standard metric up to a multiple of constant. Note that when $(M,g)$ is locally conformally flat, this functional reduces to the well-studied $\int_M \sigma_3(g)dv_g$. Hence, our result generalize a previous result of Jeff Viaclovsky without the locally conformally flat restraint.


Introduction
In the following, we let (M n , g) denote a compact, connected, smooth Riemannian manifold without boundary. We denote the Ricci curvature and scalar curvature by Ric and R, respectively. Recall that the Schouten tensor P ij is defined by and the Riemann curvature tensor can be written by where W is the Weyl curvature and ⊙ is the Kulkarni-Nomizu product, which is defined by (α ⊙ β) ijkl = α ik β jl + α jl β ik − α il β jk − α jk β il , ∀ symmetric 2-tensors α, β.
The σ k (g) curvature is defined to be the k-th elementary symmetric polynomial of the eigenvalue of the Schouten tensor P . In [V], Viaclovsky started study of the variational problems of the functional M σ k (g)dv g , he proved that the first variation of the functional M σ k (g)dv g (k = 1, 2) within a conformal class subject to the constraint V ol(M, g) = 1 is a metric satisfying σ k (g) ≡ const, and if k ≥ 3 and the Riemannian manifold is locally conformally flat, the same result follows. However, for k ≥ 3 and the manifold is not locally conformally flat, σ k (g) ≡ const is not Euler-Lagrange equation of the functional M σ k dv within a conformal class subject to the constraint V ol(M, g) = 1. The renormalized volume coefficients of g, denoted here by v (2k) (g), arose in the late 90s in the physics literature. They are defined in terms of the expansion of the ambient or Poincare metric associated to g. If the Riemannian manifold is locally conformally flat, these quantities coincide with the σ k (g) up to a constant. More precisely, it is known that (see [GJ], [CF], or [GHL]) v (2) (g) = − 1 2 σ 1 (g), v (4) (g) = 1 4 σ 2 (g), v (6) (g) = − 1 8 σ 3 (g) + 1 3(n − 4) is the Bach tensor of the metric. Just as M σ k (g −1 • A g ) dv g is conformally invariant when 2k = n and (M, g) is locally conformally flat, Graham showed in [G] that M v (2k) (g) dv g is also conformally invariant on a general manifold when 2k = n. Chang and Fang showed in [CF] that, for n = 2k, the Euler-Lagrange equations for the functional M v (2k) (g) dv g under conformal variations subject to the constraint V ol g (M ) = 1 satisfies v (2k) (g) = const., which is a generalized characterization for the curvatures σ k (g −1 • A g ) when (M, g) is locally conformally flat, as given by Viaclovsky [V]. We note that Graham [G] also gives an explicit expression of v (8) (g), but the explicit expression of v (2k) (g) for general k is not known because they are algebraically complicated (see page 1958 of [G]). Thus the study of the v (2k) (g) curvatures involves significant challenges not shared by that of σ k (g): firstly, for k ≥ 3, v (2k) (g) depends on derivatives of curvature of gin fact, for k ≥ 3, v (2k) (g) depends on derivatives of curvatures of order up to 2k − 4; secondly, the v (2k) (g) are defined via an indirect highly nonlinear inductive algorithm (see [G]). We aim to study the stability of the critical metric of the functional (n−6)/n , within a conformal class. First we recall the theorem of Chang-Fang [CF] (also see Graham [G]).
Theorem 1.1. ( [CF]) Let (M n , g) be an n-dimensional (n ≥ 7) compact Riemannian manifold, then the functional F 3 (g) is variational within the conformal class, i.e. the critical metric in [g] satisfies the equation v (6) ≡ const. (1.2) In this note, we compute the second variational formula of F 3 [g] within its conformal class [g]. Our results are Theorem 1.2. Let (M n , g) be an n-dimensional (n ≥ 7) compact Riemannian manifold with v (6) (g) =const, then the second variational formula of the functional F 3 [g] within its conformal class at g is where g t = e 2ut g, ∂ ∂t t=0 u t = φ, andφ = φ − M φdvg M dvg , T 2ij and C ijk are defined in section 2.
Theorem 1.3. Let (M n , g) be an n-dimensional (n ≥ 7) compact Einstein manifold with positive scalar curvature. Then it is a strict local maximum within its conformal class [g], unless (M n , g) is isometric to S n with the standard metric up to a multiple of constant.
Remark 1.1. When (M n , g) is a locally conformally flat, v (6) (g) = − 1 8 σ 3 (g), for the functional M σ 3 (g)dv g , J. Viaclovsky ( [V]) proved that a positive constant sectional curvature metric is a strict local minimum, unless the manifold is isometric to S n with the standard metric. Our result coincides with his at the locally conformally flat Einstein metrics, however, ours does not need the locally conformally flat assumption.

Preliminaries
Let (M n , g) be an n-dimensional compact Riemannian manifold. Throughout this note, we make the convention that repeated index means summation over 1 to n. First we recall the transformation law of various curvatures under conformal change of metrics. Letg = e 2u g, u ∈ C ∞ (M ), then the Riemannian curvature tensors satisfy where α ij = u ij − u i u j + |∇u| 2 2 g ij (note that u ij means the covariant derivative with respect to the fixed metric g). By contracting, we see that the Ricci curvature and scalar curvature satisfy (2.1) From (2.1) and the definition of Schouten tensor, we see that where we denote P (g) byP for notations convenience.
Lemma 2.1. We have the following formulae (see e.g. [GHL]) (1) ∇ i W ijkl = −(n − 3)C jkl , C ijk is the Cotton tensor defined by P ij,k − P ik,j ; (2) ∇ j B ij = (n − 4) k,l P kl C kli ; ( The proof of Lemma 2.1 is a direct calculation and one can find it in [GHL]. where I is the identity map and σ k (A) is the k-th elementary symmetric polynomial of the eigenvalues of A. Under an orthnormal basis of V , T k can be written as follows: where δ j 1 ...j k j i 1 ...i k i is the generalized Kronecker notation. We recall some well-known results in this respect, which we will need in our later arguments.
Lemma 2.2. The Newton transformations T k satisfy ( [R], [GHL]) (1) Newton's formula: In the following we denote T k (g −1 • P ) simply by T k . We have the following formula, which is a direct calculation (see [GHL] The first variational formula and proof of Theorem 1.1 In this section, we will compute the Euler-Lagrange equation for the functional F 3 (g) within the conformal class. For convenience we denote the numerator of F 3 (g) by Under the conformal change of metrics g t = e 2u(t) g, by use of (2.2), we see that in local coordinates (see [CF] For notions convenience we denote d dt by δ. Denote ∂ ∂t t=0 u = φ, and ∂ 2 ∂t 2 t=0 u = ψ. With the above preparations, we have Now we derive the first variation formula for F 3 . First we have On the other hand, the second term of (3.1) is From calculations in (3.2) and (3.3), we have the following formula, which will be used in section 4 Thus we have Proof of Theorem 1.1 Noting that u(0) = 0, we conclude the first variational formula of F 3 [g t ] within the conformal class [g] is (see [CF] or [G]) where we have used (2.3) and (2)  In this section, we will calculate the second variational formula for the functional F 3 within the conformal class [g]. The computation is direct and routine. For convenience, we separate each term in the first variational equation (3.5) and compute them respectively.
For derivative of the first term in (3.5), by use of (3.4), we have For derivative of the second term in (3.5), we need the following formula of the variation of the Newton transformation: Therefore, the variation of the second term of (3.5) is given by The variation of the third term of (3.5) is The variation of the fourth term of (3.5) is Combining (4.1), (4.3), (4.4) and (4.5), we have where we have used the following identity in the second equality which can be checked by use of (2.3), (2) of Lemma 2.1 and integration by parts. Substituting (4.7) into (4.6) and making some cancelations, we conclude that where we have used the identity that T 2kk = (n − 2)σ 2 (g) (see Lemma 2.2). It remains to study the last four terms on the right hand side of (4.8). By definition, We compute by use of divergence theorem δ mnj kli P km φ ln φ ij = φ δ mnj kli (P mk φ nl ) ,ij = φδ mnj kli P km,ij φ nl + 2P km,i φ nl,j + P km φ nl,ij . (4.9) Now we compute integrands of the right hand side of (4.9) respectively. The first term is φδ mnj kli P km,ij φ nl (4.10) The second one is The third one is φδ mnj kli P km φ nl,ij (4.12) =φφ kk,ii P nn − φφ iikl P kl − φφ klii P kl + φφ ilki P kl + φφ kiil P kl − φφ kiik P ll where we have used the Ricci identity in the last equality. Substituting the following identities into (4.11) and (4.12), after making some cancelations we see that the left hand side of (4.9) becomes φδ mnj kli P km φ nl φ ij (4.13) On the other hand, by divergence theorem, we see that the other three terms on the last of (4.8) are − (n − 2)|∇φ| 2 σ 2 (g) (4.16) = (n − 2)φ∆φσ 2 (g) + (n − 2)φφ i (σ 2 (g)) ,i By combining equations (4.13), (4.14), (4.15) and (4.16) and doing some cancelations, we conclude that the last four terms on the right hand side of (4.8) are equal to where we have used T 2ij = σ 2 (g)δ ij − σ 1 (g)P ij + P ik P kj and B ij = C ijk,k + P kl W ikjl . Moreover, where we used i C iik = 0 and C ijk = −C ikj . Thus it follows that (4.17) is equal to Substituting (4.18) into (4.8), we conclude that Proof of Theorem 1.2 By Theorem 1.1, at the critical metric of the functional F 3 (g), it holds that v (6) (g) should be constant, and it follows that F (g) = V v (6) (g). By our notations F 3 [g t ] = F (gt) ( dvg t ) (n−6)/n . By use of (4.19) and (3.6), at the critical metric g, we have If we define an operator L by for f ∈ C ∞ (M ). It is easy to see that L is self-adjoint with respect to the L 2 inner product of the Riemannian manifold. Indeed, for any two smooth functions f and h, we have where we have used (2)and (3) in Lemma 2.1, (2.3) and integration by parts. Denote φ−V −1 φ byφ. From (4.20), we see that Thus we complete the proof of Theorem 1.2.
To prove Theorem 1.3, we need the following famous theorem.
Theorem 4.1 (Lichnerowicz and Obata, see e.g. [L] Note that for an Einstein manifold, v (6) (g) = − (n−2)R 3 386n 2 (n−1) 2 , L(φ) = (n−2)R 2 64n 2 (n−1) ∆φ. Hence, we see that Therefore, we prove that an Einstein manifold with positive scalar curvature must be a strict local maximum "point" within its conformal class [g] unless (M, g) is isometric to S n with a multiple of the standard metric. We complete the proof of theorem 1.3.
Remark 4.1. Let (M n , g) be an n-dimensional Einstein manifold with nonpositive scalar curvature, then we have from the proof of Theorem 1.3 (see (4.24)) d 2 dt 2 t=0 F 3 (g t ) ≤ 0, that is, it is stable.
Remark 4.2. When M n is an Einstein manifold with positive scalar curvature with dimension n = 5, we see from (4.24) that with equality if and only if λ 1 = R 4 . Theorem 4.1 shows that in this case (M 5 , g) is isometric to the sphere S 5 with the standard metric up to a multiple of constant. And we see that this Einstein metric is a strict local minimum of the functional F 3 within its conformal class if the equality does not hold in (4.25).