A Note on the Closed Range of ∂̄_b on q-Convex Manifolds

Theorem 1.1. Let X be a complex manifold and $M \subset \subset X$ a smooth compact, pseduconvex-oriented CR submanifold of hypersurface type of dimension 2p-1. Assume that there exists a (q+1)-convex function $\phi$ defined in a neighborhood of M in X. Then $$\bar{\partial}_b \colon Dom(\bar{\partial}_b)_{r,s-1} \longrightarrow L^2_{r,s}(M)$$ has closed range if $n-q \le s \le p+q-n$ and p>2(n-q). Moreover, if X is completely q-convex, then the same conclusion holds for p=2(n-q). The paper is organized as follows. In Section 2 recall some basic definitions and we show how to realize a CR manifold of hypersurface type as the boundary of a complex manifold Y. In Section 3 we present the proof of Theorem 1.1. This paper was written for a special volume of CASJ dedicated to the memory of Nicholas Hanges. He did pioniering work on propagation of holomorphic extendibility of CR functions, and we use related results in this paper. We will remember him as a prominent mathematician, a wonderful person, and a great colleague.

Proposition 2.4. Let X be a complex manifold and M a smooth, compact, connected CR submanifold of hypersurface type. Let $\phi$ be a (q+1)-convex function defined on a neighborhood of M in X. Assume that the dimension of M is 2p-1, with p > n-q. Then M consists of a single CR orbit.

Proposition 2.5. Let $M \subset\subset X$ be a smooth, compact, connected, pseudoconvexoriented CR manifold of hypersurface type of dimension 2p-1. Let $\phi$ be a (q+1)convex function defined on a neighborhood of M in X, with q>n-p. Then Mis endowed with a partial one-sided complexification in X. That is, there exists a complex manifold $Y\subset\subset X$ which has M as the smooth connected component of its boundary on the pseudoconvex side.

Lemma 2.6. Let $M \subset\subset X$ be a smooth, pseudoconvex-oriented CR manifold of hypersurface type. Let $\gamma$ be a piecewise smooth CR curve connecting two points $z_0$ and $z_1$ of M. If M has complex extension in direction $+JiT(z_0)$ at $z_0$ , then M also has complex extension in direction $+JiT(z_1)$ at $z_1$ . Remark 2.7. In the proof of Proposition 2.4 we have used [20] and [21] to build a one sided complexification of M near minimal points. The results in [20] and [21], however, do not specify on which side of M this complexification lies. Our hypotheses on the pseudoconvexity of M assures that the side of the complexification at minimal points is the pseudoconvex side of M, namely the side pointed by JiT.

Proposition 3.1. Let M and $\phi$ be as in Proposition 2.5 and assume further that p > 2(n-q). Then there exists a complex sub-manifold Y of X of dimension p with smooth boundary $\partial Y$ such that $\partial Y = M \cup M_2$ , where $M_2$ is CR of hypersurface type. Moreover, if $\bar{\partial}$ is the Cauchy Riemann operator on Y, for a suitable weight function $\lambda$ we have, for all $f \in Dom(\bar{\partial}_t^*)_{r,s} \cap C_{(r,s)}^{\infty}(\overline{Y})$ , that <span id="page-5-1"></span> $$t\|f\|_t^2 \leq C(\|\bar{\partial}f\|_t^2 + \|\bar{\partial}_t^*f\|_t^2) + C_t\|f\|_{-1}^2 \quad \forall s \geq n-q, \ s \leq p+q-n-1, \ \forall t > 0. \ (3.1)$$

Definition 2.1. We say that $\phi$ is q-convex if the Levi form $\partial \bar{\partial} \phi$ has at least q positive eigenvalues.

Definition 2.2. We say that X is completely q-convex if there exists a smooth exhaustion function $\phi: X \to \mathbb{R}$ which is (q+1)-convex. Let $J: TX \to TX$ be the standard complex structure induced by the multiplication by i and let M be a real submanifold of X.

Definition 2.3. The complex tangent space to M at a point $z \in M$ is the subspace $$T_z^{\mathbb{C}}M:=T_zM\cap JT_zM.$$ We will say that M is a CR manifold if $T_z^{\mathbb{C}}M$ has constant dimension. The bundle so formed is called the complex tangent bundle of M and is denoted by $T^{\mathbb{C}}M$ . We say that M is of hypersurface type if $\frac{TM}{T^{\mathbb{C}}M}$ has rank 1. Let M be a smooth, compact CR submanifold of X equipped with the induced CR structure $T^{1,0}M = \mathbb{C}TM \cap T^{1,0}X$ . The De Rham exterior derivative induces a complex on skew-symmetric antiholomorphic forms on M. We denote such complex by $\bar{\partial}_b$ . Assume that M is of hypersurface type. Hence the complexified tangent bundle $\mathbb{C}TM$ is spanned by $T^{1,0}M$ , its conjugate $T^{0,1}M$ and a single additional vector field T. We can assume T to be purely imaginary, that is, satisfying $\overline{T} = -T$ . Let $\eta$ be a purely imaginary 1-form which annihilates $T^{1,0}M \oplus T^{0,1}M$ and normalized so that $\langle \eta, T \rangle = -1$ . The manifold M is orientable if there exists a global 1-form section $\eta$ (or vector field T) and is pseudoconvex if the hermitian form defined on $T^{1,0}M$ by $d\eta(X,Y)=\langle d\eta,X\wedge\overline{Y}\rangle$ is positive semidefinite. We say that M is pseudoconvex-oriented if both properties are satisfied at the same time. A CR curve $\gamma$ on M is a real curve such that $T\gamma \subset T^{\mathbb{C}}M$ . A CR orbit is the union of all piecewise smooth CR curves issued from a point of M. We denote by $\mathcal{O}(z)$ the CR orbit of a point $z \in M$ , and we say that a set S is CR invariant if $\mathcal{O}(z) \subset S$ for all $z \in S$ . By Sussmann's Theorem [14], the orbit $\mathcal{O}(z)$ has the structure of an immersed variety of X. Following [2] and [8], we prove that the manifold M in question consists of a single orbit. The difference with [2, 8] is that instead of holomorphic coordinate functions that are not available here, we use the given (q+1)-convex function.