Pseudo-Einstein and Q-flat Metrics with Eigenvalue Estimates on CR-hypersurfaces

In this paper, we will use the Kohn’s ∂̄b-theory on CRhypersurfaces to derive some new results in CR-geometry. Main Theorem. LetM2n−1 be the smooth boundary of a bounded strongly pseudo-convex domain Ω in a complete Stein manifold V 2n. Then: (1) For n ≥ 3, M2n−1 admits a pseudo-Einstein metric. (2) Forn ≥ 2,M2n−1 admits a Fefferman metric of zero CRQ-curvature. (3) In addition, for a compact strictly pseudoconvex CR emendable 3manifold M3, its CR Paneitz operator P is a closed operator. There are examples of non-emendable strongly pseudoconvex CRmanifolds M3, for which the corresponding ∂̄b-operator and Paneitz operators are not closed operators. 0. INTRODUCTION In this paper, we study several questions, including the existence ofQ-flat metrics, pseudo-Einstein metrics, and the closedness of the CR Paneitz operators. First, we will use an approach proposed by Fefferman and his school to prove that “the complete Kähler-Einstein g∞ on an open domain Ω induces a metric on M = bΩ with zero CR Q-curvature, where Ω is a smooth, bounded strictly pseudo-convex domain in a Stein manifold V 2n.” To achieve this goal, we solve a ∂∂̄-Poincaré-LeLong equation via the ∂̄-theory. Although this part does not produce new hard a-priori estimates, it is still valuable for other potential applications. The second purpose is to prove the existence of pseudo-Einstein metrics on strictly pseudo-convex CR-hypersurface of real dimension ≥ 5 through solving the ∂̄b Poincaré-LeLong equations. The last part of our paper is to study the closedness of CR Paneitz operator, which is a fourth-order differential operator. It is known that the positivity 2839 Indiana University Mathematics Journal c ©, Vol. 56, No. 6 (2007) 2840 JIANGUO CAO & SHU-CHENG CHANG of a CR Paneitz operator is related to the deformation of Q-curvatures under the conformal change of metrics on Riemannian manifold Mm. In particular, the positivity of a CR Paneitz operator is also related to the lower bound of the first eigenvalue of sub-Laplace on a CR-manifold M3, see [5], [6] and [25]. It will be shown that, if M3 = bΩ4 is the smooth boundary of a bounded strictly pseudoconvex domain Ω in a Stein manifold V 4, then its CR Paneitz operator on M3 is closed, for any metric on M3. Main Theorem. Let M2n−1 be the smooth boundary of a bounded strongly pseudo-convex domain Ω in a complete Stein manifold V 2n. Then (1) For n ≥ 2, M2n−1 admits a metric of zero CR Q-curvature; (2) For n ≥ 3, M2n−1 admits a pseudo-Einstein metric; (3) In addition, for a compact strictly pseudoconvex CR emendable 3-manifold M3, its CR Paneitz operator P is a closed operator. Earlier work in this direction for the case of V 2n = Cn can be found in [24], [15] and [16]. In a very recent paper [25], Li and Luk obtained an explicit formula for Webster’s pseudo-Ricci curvature on real hypersurfaces in Cn. Thus, their result could lead another proof of Cheng-Yau’s result ([8]) and Mok-Yau’s theorem [27], which will be used in Section 2 below. Among other things, we introduce some new methods to handle pseudoEinstein metric and Paneitz operators in this paper. For example, we use the closeness of ∂̄b and ∂̄∗ b operators provided by Kohn’s theory, in order to complete the proof. When dimR[M] = 3, we decompose the Paneitz operator P as a product of closed operators. Thus, the closed property of P will follow immediately, see Lemma 1.4 and Section 4 below. 1. PRELIMINARY RESULTS It is well known that the real Laplace ∆ on a Kähler manifold V 2n satisfies ∆ = 2 = 2̄, where = ∂̄∂̄∗ + ∂̄∗∂̄ is the complex Laplace operator. However, it may happen that ∆b ≠ 2 b in some cases. Let us recall the notions of ∆b and b. Since M2n−1 = bV 2n has odd real dimension, it is a Cauchy-Riemann manifold. The ∂̄b operator induces a sub-elliptic operator b = ∂̄∗ b ∂̄b + ∂̄b∂̄∗ b acting on L(p,q)(M). Similarly, there is a real sub-Laplace operator, which can be viewed as a partial trace of the hessian operator (or can be viewed as a sum of the squares of (2n− 2) vectors): ∆bu∣∣z = 2n−2 ∑ k=1 〈∇ek(∇u), ek〉 ∣∣∣ z Pseudo-Einstein and Q-flat Metrics 2841 where e2n is the outward real unit normal vector of Ω along boundary M = bΩ, e2j = Je2j−1 for j = 1, . . . , n, z ∈ M, J is the complex structure of V 2n, {e1, e2, . . . , e2n−3, e2n−2, e2n−1, e2n} is an orthonormal basis of [Tz(V)]R, and