Some variational problems in conformal geometry

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228 B. GUO AND H. LI First we recall Theorem 6.1 of Chang-Fang states that the critical metric g w of F 3 in the conformal class [g] satisfies the equation (6.4) v (6) (g w ) ≡ const. The second variational formula of F 3 [g] within the conformal class [g] is computed by Guo-Li ([GL1]). Their results are as follows: Theorem 6.2 ([GL1]). 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 t ] within its conformal class at g is (6.5) { ∣ d 2 ∣∣t=0 ∫ [ dt F 2 3 [g t ] = (n − 6)V −(n−6)/n − 6v (6) (g) ¯φ 2 where g t = e 2u t ∂ g, ∣ t=0 u t = φ, and ¯φ = φ − ∂t ( Bij ¯φij + 24(n−4) + 1 8 T ¯φ ) ] 2ij ij − 1 12 A ijC ijk ¯φk ¯φ dv ∫ M φdv g ∫M dv , g (6.6) T 2ij = σ 2 g ij − σ 1 A ij + A ik A kj is one of the Newton transformations associated to A ij , and the Cotton tensor is defined in (1.3), V = ∫ M dv g is the volume of (M,g). From the second variational formula of F 3 , and by using Lichnerowicz and Obata theorem on the lower bound of the first nonzero eigenvalue of the Laplacian on compact manifolds with positive Ricci curvature lower bound, we proved the following stability result: Theorem 6.3 ([GL1]). 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 some multiple constant. ∫ Remark 6.4. When (M n ,g) is locally conformally flat, for the functional M σ 3(g)dv g , J. Viaclovsky ([V1]) proved that a metric with positive constant sectional curvature is a strict local minimum, unless the manifold is isometric to S n with the standard metric. Theorem 6.3 coincides with his at the locally conformally flat Einstein metrics, however, it does not need the assumption of locally conformally flatness in Theorem 6.3. Remark 6.5. Let (M n ,g)beann-dimensional Einstein manifold with nonpositive scalar curvature, from second variational formula Theorem 6.2, we have that d 2 ∣ ∣∣t=0 dt 2 F 3 (g t ) ≤ 0. Remark 6.6. Recently, by using a first variational formula of Graham, A. Chang, H. Fang and M. Graham simplify the computation of the second variational formula for F 3 , and they can also give the second variational formula for general functionals F k and obtain similar stability result for Einstein manifolds as in Theorem 6.3. } This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
SOME VARIATIONAL PROBLEMS IN CONFORMAL GEOMETRY 229 For the general first variational formula of F 3 (g) = ∫ M v(6) (g)dv g in arbitrary direction h ij ,whereh ij is any symmetric 2-tensor, Fang-Guo-Li ([FGL]) obtained the following formula d ∫ ∫ (6.7) ∣ v (6) (g t )dv gt = F ij h ij dv, dt t=0 M ∣ d where g t is a family of Riemannian metrics, dt (g ∣∣t=0 t) ij symmetric 2-tensor defined by = h ij , and F ij is a (6.8) F ij = − ΔB ij 3(n−2)(n−4) − C iklC jkl 6(n−4) − 2A klC (ij)k,l 3(n−4) − ∇ kσ 1 C (ij)k 3(n−4) + C kilC ljk 3(n−4) − 2B klW ikjl 3(n−4)(n−2) + 1 3(n−4) σ 1B ij − 2 3(n−2) A kmA ml W ikjl +(n − 6)[− 1 6(n−4)(n−2) M (B ik A kj + B jk A ki ) − 1 6(n−1)(n−4) (σ 2) ij − 1 6(n−2)(n−4) ΔT 2 ij + 1 + 1 2(n−2) v(6) g ij 6(n−1)(n−4) Δσ 2g ( ij 1 + 3(n−4)(n−2) σ 1A kl W ikjl − 1 3(n−2) T ik 2 A n kj + 6(n−1)(n−4) σ 2 A ij − σ 1 n g ij )], where B ij =(B g ) ij is the Bach tensor defined ) in (4.6), C ijk is the Cotton tensor defined in (1.3), and C (ij)k := 1 2 (C ijk + C jik is the symmetrization of the tensor C ijk with respect to the first two indices. Remark 6.7. When n = 6, the 2-tensor F ij defined in (6.8) coincides with the “obstruction tensor” O ij defined in [GrH], precisely, O ij = −24F ij =ΔB ij +2C ikl C jkl +8A kl C (ij)k,l +4∇ k σ 1 C (ij)k − 4C kil C ljk − 2B kl W kijl − 4σ 1 B ij +4A km A ml W ikjl . We note that O ij has the following important properties when n =6: (1) O ij is trace-free and divergence-free, i.e., ∑ ∑ O ii =0, O ij,j =0. i (2) O ij is conformally invariant of weight 2 − n; i.e., if 0
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Some variational problems in conformal geometry

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