Some variational problems in conformal geometry

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224 B. GUO AND H. LI where a is a nonpositive constant. In [GL2], the authors proved that under some assumptions, the critical metric to the functional F a is also of constant curvature, more precisely, they showed the following result: Theorem 2.5. Let (M 3 ,g) be a compact three dimensional Riemannian manifold. Then a critical metric g to F a | M1 satisfies (2.8) b := σ 2 + aσ 1 ≡ constant. Moreover, if b+ 3 4 a2 > 0, the critical metric must be of constant sectional curvature. Remark 2.6. The analogue of Theorem 2.5 holds for higher dimensional locally conformally flat manifolds. 3. Locally conformally flat manifolds with σ k (g) =constant In this section, we will study the global properties of a locally conformally flat Riemannian manifold (M n ,g) with constant σ k (g) forsomek ∈{1, 2,...,n}. 1 Since σ 1 (g) = 2(n−1) R, σ 1(g) is constant if and only if (M,g) hasconstant scalar curvature R. On the other hand, from the solution of Yamabe problem, we know that every compact conformally flat manifold (M,g) admits a pointwise conformal metric ˜g such that (M n , ˜g) is a conformally flat manifold with constant scalar curvature. The following theorem is well known. Theorem 3.1 ([Ta], [We], [Ch]). Let (M n ,g) be a compact conformally flat Riemannian manifold with constant scalar curvature. If the Ricci tensor is semipositive definite, then (M n ,g) is either a space form, or a space S 1 × N n−1 ,where N n−1 is an (n − 1)- dimensional space form. On the other hand, for complete conformally flat manifolds with non-negative Ricci curvature, S. H. Zhu ([Zh]) established the following theorem. Theorem 3.2 ([Zh]). If (M n ,g) is a complete locally conformally flat Riemannian manifold with Ric(g) ≥ 0, then the universal cover ˜M of M with the pullback metric is either conformally equivalent to S n , R n or is isometric to R×S n−1 (c). If M n itself is compact, then ˜M is either conformally equivalent to S n (1) or isometric to R n , R × S n−1 (c). Motivated by Theorems 3.1 and 3.2, one may try to classifiy compact locally conformally flat Riemannian manifolds with constant σ k (g) (k ≥ 2). In this respect, Z.Hu,H.LiandU.Simon([HLS]) proved that Theorem 3.3 ([HLS]). Let (M n ,g) be a compact locally conformally flat manifold with constant non-zero σ k (g) for some k ∈{2, 3,...,n}. IfthetensorA g is semi-positive definite, then (M n ,g) is a space form of positive sectional curvature. In [GVW], Guan-Viaclovsky-Wang established the following classification result: Theorem 3.4 ([GVW]). Let (M n ,g) be a compact locally conformally flat manifold with nonnegative σ k (g) for some k ≥ n 2 .Then(M n ,g) is conformally equivalent to either a space form or a finite quotient of a Riemannian product S n−1 (c) × S 1 for some constant c>0 and k = n 2 .Inparticular,ifg ∈ Γ+ k ,then (M n ,g) is conformally equivalent to a spherical space form. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
SOME VARIATIONAL PROBLEMS IN CONFORMAL GEOMETRY 225 If A g is semi-positive definite, then (M n ,g) has nonnegative scalar curvature and nonnegative Ricci curvature. For the case k =2,Z.Hu,H.LiandU.Simon ([HLS]) showed that Theorem 3.5 ([HLS]). Let (M n ,g) be a compact locally conformally flat manifold with σ 2 (g) a non-negative constant. If the Ricci tensor is semi-positive definite, then (M n ,g) is a space form for n =3and is either a space form or a space S 1 × N n−1 with N n−1 a space form for n ≥ 4. For the complete case, similar results are rare. Z. Hu, H. Li and U. Simon proved in [HLS]: Theorem 3.6 ([HLS]). Let (M n ,g) be a complete locally conformally flat manifold with constant scalar curvature. If the Ricci tensor is semi-positive definite, then the universal cover ˜M of M n with pull-back metric is isometric to S n (c), R n or R × S n−1 (c). 4. Renormalized volume coefficients v (2k) (g) Renormalized volume coefficients, v (2k) (g), of a Riemannian metric g, were introduced in the physics literature in the late 1990’s in the context of AdS/CFT correspondence. Given a manifold (X n+1 ,M n ,g + ) with boundary ∂X = M. Letr be a defining function for M n in X n+1 such that r>0inX and r =0onM, while dr| M ≠0. For any given Riemannian manifold (M n ,h 0 ), Fefferman-Graham ([FG]) proved that there is an extension g + of h 0 , which is “asymptotically Poincare Einstein” in a neighborhood of M n , i.e., in [0,ɛ) × M n for some ɛ>0, (4.1) Ric(g + )=−ng + as r → 0. Locally we can write (4.2) g + = 1 r 2 (dr2 + h ij (r, x)dx i dx j ), where h(0, ·) =h 0 (·) andh(r, ·) is a metric defined on M c := {r = c} ⊂X when c is sufficiently small, {x i } are local √ coordinates of M n . det hij (r,x) Suppose the expression of √ near r = 0 is given by det hij (0,x) √ det hij (r, x) ∞ (4.3) √ det hij (0,x) = ∑ v (k) (x, h 0 )r k , where v (k) (x, h 0 ) is a curvature invariant of the metric h 0 for 2k ≤ n. The followings are some important properties of v (k) (g): • Graham ([G0]): v (k) vanishes for k odd. • Graham ([GJ]): when (M,g) is locally conformally flat, v (2k) (g) isequal to σ k (g) up to a scaling constant. Here are some expressions of v (k) (g) for lower k’s. k=0 (4.4) v (2) (g) =− 1 2 σ 1(g), v (4) (g) = 1 4 σ 2(g). For k = 3, Graham and Juhl ([GJ]) listed the following formula for v (6) (g): (4.5) v (6) (g) =− 1 8 [σ 1 3(g)+ 3(n − 4) (A g) ij (B g ) ij ], This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
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