Connected sum of CR manifolds with positive CR Yamabe constant

TL;DR

This paper proves that the connected sum of two 3-dimensional closed CR manifolds with positive CR Yamabe constants also admits a CR structure with a positive CR Yamabe constant, extending the class of such manifolds.

Contribution

It demonstrates that the property of having a positive CR Yamabe constant is preserved under the connected sum operation for 3-dimensional CR manifolds.

Findings

Connected sum of two CR manifolds with positive CR Yamabe constant also has positive CR Yamabe constant.

Extends the class of CR manifolds known to admit positive CR Yamabe structures.

Provides a method to construct new CR manifolds with positive CR Yamabe constant from existing ones.

Abstract

Suppose $M_{1}$ and $M_{2}$ are $3$-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a CR structure with positive CR Yamabe constant.

Equations

$$L_{\theta }(X,Y)=2d\theta (X,JY),X,Y\in \xi .$$

$$Xu=L_{\theta }(X,{\nabla }_{b}u),$$

$$|{\nabla }_{b}u{|}_{\theta }^2=L_{\theta }({\nabla }_{b}u,{\nabla }_{b}u).$$

$$L_{\theta }({\nabla }_{b}u,{\nabla }_{b}v)=L_{\theta }^{\ast }(du,dv)=L_{\theta }^{\ast }(d_{b}u,d_{b}v),$$

$${\int }_M({\Delta }_{b}u)vdV={\int }_M{L}_{\theta }^{\ast }(du,dv)dV,$$

$$\tilde{R}=u^{1-p}(b_n{\Delta }_b+R)u$$

$$\lambda (M,J)=\underset{u>0}{\inf }\frac{E_{\theta }(u)}{({\int }_M{u}^{2+\frac{2}{n}}d{V}_{\theta }{)}^{\frac{n}{n+1}}},$$

$$E_{\theta }(u)={\int }_M\left((2+\frac{2}{n})|{\nabla }_{b}u{|}^2+R{u}^2\right)d{V}_{\theta }.$$

$$\begin{array}{rl}d{\theta }^1& ={\theta }^1\wedge {\theta }_1{}^1+\theta \wedge {\tau }^1,\\ 0& ={\theta }_1{}^1+{\theta }_{\bar{1}}{}^{\bar{1}},\\ 0& ={\tau }_1\wedge {\theta }^1,\end{array}$$

$$(x_1,y_1,z_1)\circ (x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2+y_1{x}_2-x_1{y}_2),$$

$${\mathring{e}}_1=\frac{\partial }{\partial x}+y\frac{\partial }{\partial z},{\mathring{e}}_2=\frac{\partial }{\partial y}-x\frac{\partial }{\partial z}\text{mbox}andT=\frac{\partial }{\partial z}.$$

$$\Theta =dz+xdy-ydx.$$

$$\mathring{J}({\mathring{e}}_1)={\mathring{e}}_2\text{mbox}andJ({\mathring{e}}_2)=-{\mathring{e}}_1.$$

$$d\Theta (X,JX)>0\text{mbox}foranynonzerovectorX\in \mathring{\xi }.$$

$$Z_1=a_1{}^1{\mathring{Z}}_1+b_1{}^{\bar{1}}{\mathring{Z}}_{\bar{1}},$$

$$Z_1\wedge Z_{\bar{1}}=(|a_1{}^1{|}^2-|b_1{}^{\bar{1}}{|}^2){\mathring{Z}}_1\wedge {\mathring{Z}}_{\bar{1}}.$$

$$|a_1{}^1{|}^2>|b_1{}^{\bar{1}}{|}^2.$$

$$\varphi =(a_{\bar{1}}{}^{\bar{1}}{)}^{-1}{b}_{\bar{1}}{}^1,$$

$$\begin{array}{rl}& \theta {|}_U=\Theta ,\\ & \varphi (0)={\varphi }_1(0)={\varphi }_{\bar{1}}(0)=0.\end{array}$$

$$Z_{\bar{1}}={\left(\frac{1}{1-|\varphi {|}^2}\right)}^{1/2}({\mathring{Z}}_{\bar{1}}+\varphi {\mathring{Z}}_1),$$

$${\theta }^1={\left(\frac{1}{1-|\varphi {|}^2}\right)}^{1/2}({\mathring{\theta }}^1-\varphi {\mathring{\theta }}^{\bar{1}}).$$

$$\begin{array}{rl}{A}^1{}_{\bar{1}}& =-\frac{{\varphi }_0}{1-|\varphi {|}^2};\\ {\theta }_1{}^1& =-d\ln {\left(\frac{1}{1-|\varphi {|}^2}\right)}^{1/2}+\left[\frac{\bar{\varphi }{\varphi }_0}{1-|\varphi {|}^2}\right]\Theta \\ & +\left[\frac{{\bar{\varphi }}_{\bar{1}}+\bar{\varphi }{\varphi }_1}{1-|\varphi {|}^2}+\bar{\varphi }{\mathring{Z}}_{\bar{1}}\left(\frac{1}{1-|\varphi {|}^2}\right)+{\mathring{Z}}_1\left(\frac{1}{1-|\varphi {|}^2}\right)\right]{\mathring{\theta }}^1\\ & -\left[\frac{{\varphi }_1+\varphi {\bar{\varphi }}_{\bar{1}}}{1-|\varphi {|}^2}+|\varphi {|}^2{\mathring{Z}}_{\bar{1}}\left(\frac{1}{1-|\varphi {|}^2}\right)+\varphi {\mathring{Z}}_1\left(\frac{1}{1-|\varphi {|}^2}\right)\right]{\mathring{\theta }}^{\bar{1}},\end{array}$$

$$\begin{array}{rl}{R}^{\varphi ,\Theta }& =-{\mathring{Z}}_{\bar{1}}\left[\frac{{\bar{\varphi }}_{\bar{1}}+\bar{\varphi }{\varphi }_1}{1-|\varphi {|}^2}+\bar{\varphi }{\mathring{Z}}_{\bar{1}}\left(\frac{1}{1-|\varphi {|}^2}\right)+{\mathring{Z}}_1\left(\frac{1}{1-|\varphi {|}^2}\right)\right]\\ & -{\mathring{Z}}_1\left[\frac{{\varphi }_1+\varphi {\bar{\varphi }}_{\bar{1}}}{1-|\varphi {|}^2}+|\varphi {|}^2{\mathring{Z}}_{\bar{1}}\left(\frac{1}{1-|\varphi {|}^2}\right)+\varphi {\mathring{Z}}_1\left(\frac{1}{1-|\varphi {|}^2}\right)\right]\\ & +\frac{i\bar{\varphi }{\varphi }_0}{1-|\varphi {|}^2},\end{array}$$

$$\begin{array}{rl}{\Delta }_b^{\varphi ,\Theta }u& =\left[\frac{1+|\varphi {|}^2}{1-|\varphi {|}^2}\right]{\mathring{\Delta }}_{b}u-\left[\frac{2\bar{\varphi }}{1-|\varphi {|}^2}\right]u_{\bar{1}\bar{1}}-\left[\frac{2\varphi }{1-|\varphi {|}^2}\right]u_{11}\\ & -\left[\frac{2{\bar{\varphi }}_{\bar{1}}+|\varphi {|}_1^2}{1-|\varphi {|}^2}+2\bar{\varphi }{\mathring{Z}}_{\bar{1}}\left(\frac{1}{1-|\varphi {|}^2}\right)+(1+|\varphi {|}^2){\mathring{Z}}_1\left(\frac{1}{1-|\varphi {|}^2}\right)\right]u_{\bar{1}}\\ & -\left[\frac{2{\varphi }_1+|\varphi {|}_{\bar{1}}^2}{1-|\varphi {|}^2}+(1+|\varphi {|}^2){\mathring{Z}}_{\bar{1}}\left(\frac{1}{1-|\varphi {|}^2}\right)+2\varphi {\mathring{Z}}_1\left(\frac{1}{1-|\varphi {|}^2}\right)\right]u_1.\end{array}$$

$$d{\theta }_1{}^1=R{\theta }^1\wedge {\theta }^{\bar{1}}\text{mod}\theta ,$$

$$\Delta_{b}^{\phi,\Theta}u=-\Big{(}Z_{\bar{1}}Z_{1}u-\theta_{1}{}^{1}(Z_{\bar{1}})Z_{1}u\Big{)}+\ \text{conjugate},$$

$$R^{\varphi ,{\theta }^u}=u^{-3}(4{\Delta }_b^{\varphi ,\Theta }u+R^{\varphi ,\Theta }u),$$

$$R^{0,{\theta }^u}(0)=R^{\varphi ,\Theta }(0).$$

$$R^{0,{\theta }^u}=u^{-3}(4{\mathring{\Delta }}_{b}u).$$

$$\begin{array}{rl}{R}^{\varphi ,\Theta }& =-F^2({\varphi }_{11}+{\bar{\varphi }}_{\bar{1}\bar{1}})+\sum _{a,b\in \{1,\bar{1}\}}{P}_{ab}{\varphi }_{ab}+\sum _{a,b\in \{1,\bar{1}\}}{Q}_{ab}{\bar{\varphi }}_{ab}+P,\\ {\Delta }_b^{\varphi ,\Theta }u& =F^2{\mathring{\Delta }}_{b}u+\sum _{a,b\in \{1,\bar{1}\}}{S}_{ab}{u}_{ab}+S{u}_1+\bar{S}{u}_{\bar{1}},\end{array}$$

Full text

1. Introduction

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. It was shown by Gromov-Lawson in [5] and independently by Schoen-Yau in [9] that the connected sum of two closed (that is, compact without boundary) manifolds of positive scalar curvature has a metric of positive scalar curvature. It was also shown by Schoen-Yau in [9] that the connected sum of two closed conformally flat manifolds of positive scalar curvature has a conformally flat metric of positive scalar curvature (see Corollary 5 in [9]). In view of the similarity between the scalar curvature in Riemannian geometry and the Tanaka-Webster scalar curvature in CR geometry, it would be natural to ask if the corresponding results hold for the Tanaka-Webster scalar curvature. It is the purpose of this note to answer this question.

For basic materials in CR geometry and pseudohermitian geometry, we refer the readers to [4], [6], [8] or [10]. and the references therein. Let $(M,J)$ be a closed, strictly pseudoconvex CR manifold of dimension $2n+1$. Take a contact form $\theta$, which means a $1$-form satisfying the complete non-integrability condition: $\theta\wedge(d\theta)^{n}\neq 0$ for each point of $M$. Let $\xi=\ker{\theta}$, which is the associated contact bundle. We choose $\theta$ compatible with $J$ in the following sense: the CR structure $J$ is defined on $\xi$ and $d\theta(X,JX)>0$ for any nonzero vector $X\in\xi$. The Levi metric (or form) of $\theta$ is the real symmetric bilinear form $L_{\theta}$ on $\xi$ defined by

[TABLE]

$L_{\theta}$ extends by complex linearity to $\xi\otimes C$, and induces a hermitian metric (or form) on the subbundle $\xi_{1,0}\subset\xi\otimes C$ of all CR holomorphic vectors (Note that, instead of $\xi_{1,0}$, D. Jerison and J. Lee in [6] used $T_{1,0}$ to denote the CR holomorphic subbundle). For a real function $u$, the subgradient of $u$ is denoted by $\nabla_{b}u$ and defined as the unique vector in $\xi$ satisfying

[TABLE]

for all $X\in\xi$. Here $Xu$ means the directional derivative of $u$ along $X$. The norm of $\nabla_{b}u$ is defined by

[TABLE]

In [4], S. Dragomir and G. Tomassini considered the gradient $\nabla u$ of $u$ with respect to the Webster metric $g_{\theta}$. It is easy to see that $\nabla_{b}u=\pi_{H}\nabla u$, where $\pi_{H}:TM\rightarrow\xi$ is the natural orthogonal projection defined in [4] (in which, instead of $\xi$, they used $H(M)$ to denote the contact bundle). It is also easy to check that

[TABLE]

for any real functions $u$ and $v$, where $L_{\theta}^{*}$ is the induced metric on $\xi^{*}$, determined by $L_{\theta}$, and extends naturally to $T^{*}M$ (see [6] for more details) and $d_{b}=\bar{\partial}_{b}+\partial_{b}$. Take the volume form $dV:=\theta\wedge(d\theta)^{n}.$ The sublaplacian operator $\Delta_{b}$ is defined on real functions $u\in C^{\infty}(M)$ by

[TABLE]

for all $v\in C^{\infty}_{0}(M)$.

If $\theta$ is replaced by $\tilde{\theta}=u^{p-2}\theta$, with $p=2+\frac{2}{n}$, then we have the transformation law for Tanaka-Webster scalar curvatures

[TABLE]

where $b_{n}=2+\frac{2}{n}$, and $R$ (or $R_{J,\theta}$) and $\tilde{R}$ (or $R_{J,\tilde{\theta}}$) are respectively Tanaka-Webster scalar curvature on the pseudohermitian manifold $(M,J,\theta)$ and $(M,J,\tilde{\theta})$.

We define the CR Yamabe constant $\lambda(M,J)$ (or $\lambda(M)$ if $J$ is clear in the context) as follows: (see [6])

[TABLE]

where

[TABLE]

Similar to the Riemannian case, one can show that $\lambda(M,J)>0$ if and only if there exists a contact form $\tilde{\theta}$ conformal to $\theta$ such that the Tanaka-Webster scalar curvature of $\tilde{\theta}$ is positive.

In [1], the first and the second authors proved the following theorem, which is the CR version of Schoen-Yau’s result mentioned above (see also [7] for a different proof by O. Kobayashi). Recall that a CR manifold is called spherical if it is locally CR isomorphic to the standard CR sphere $S^{2n+1}$.

Theorem 1.1**.**

$($[1]$)$* Suppose $(M_{1},J_{1})$ and $(M_{2},J_{2})$ are two closed, spherical CR manifolds of dimension $2n+1$ with $\lambda(M_{k},J_{k})>0$ for $k=1,2$. Then their connected sum $M_{1}\#M_{2}$ admits a spherical CR structure $J$ with $\lambda(M_{1}\#M_{2},J)>0$.*

The idea of the proof of Theorem 1.1 was motivated by the work of O. Kobayashi in [7]. More precisely, we fix a point $p_{j}\in M_{j}$ for $j=1,2$. We first take off two small balls around $p_{1}$ and $p_{2}$. Since $M_{j}$ are spherical, we can attach the Heisenberg cylinder in each of punched neighborhood of $p_{j}$. We then glue two Heisenberg cylinders together to get a spherical CR manifold.

In this note, we continue our study on the Tanaka-Webster scalar curvature of connected sum on CR manifolds without assuming they are spherical. In particular, we prove the following theorem, which can be viewed as the analogous result of Gromov-Lawson and of Schoen-Yau mentioned above.

Theorem A. Suppose $(M_{1},J_{1})$ and $(M_{2},J_{2})$ are two $3$-dimensional closed CR manifolds with $\lambda(M_{k},J_{k})>0$ for $k=1,2$. Then their connected sum $M_{1}\#M_{2}$ admits a CR structure $J$ with $\lambda(M_{1}\#M_{2},J)>0$.

Note that the above argument for the spherical case cannot be applied directly, since we cannot attach the Heisenberg cylinder to the punched neighborhood of a point. However, in this paper, we will mainly construct a new CR structure, through the deformation tensor, which outside a ball is the given CR structure and is spherical in a neighborhood contained in the ball. In addition, we can construct such a CR structure such that its Yamabe constant is as close as possible to the one of the given CR structure. Hence, together with Theorem 1.1, we obtain Theorem A. The idea of the proof is elegant and also, as Theorem 1.1, motivated by the work of Kobayashi in [7]. However, due to different nature of geometric structures, the way to construct the new CR structure is entirely different from that in Riemannian geometry. We use the concept of the deformation tensor, which has been well studied in $3$-dimensional CR geometry.

We learned that Dietrich [3], among others, also claimed the same statement as in Theorem A. But the proof of his key lemma (Lemma 5.6 in [3]) is not clear to us (cf. the proof of Proposition 3.6 in this paper).

**Acknowledgments. **J.-H. Cheng (resp. H.-L. Chiu) would like to thank the Ministry of Science and Technology of Taiwan, R.O.C. for the support of the project: MOST 107-2115-M-001-011- (resp. MOST 106-2115-M-007-017-MY3). J.-H. Cheng would also like to thank the National Center for Theoretical Sciences for the constant support.

2. Basic Material

For basic material in $CR$ and pseudohermitian geometry, we refer the reader to [4], [6], [8] or [10]. Let $(M^{3},J,\theta)$ be a pseudohermitian manifold. In [10], S. Webster showed that there is a natural connection in the bundle $\xi_{1,0}$ of all CR holomorphic vectors adapted to the pseudohermitian structure $(J,\theta)$. To define the connection, choose an orthonormal admissible coframe $\{\theta^{1}\}$ and dual frame $\{Z_{1}\}$ for $\xi_{1,0}$. Webster showed that there are uniquely determined $1$-forms $\theta_{1}{}^{1},\tau^{1}$ on $M$ satisfying the following structure equations

[TABLE]

in which $\tau_{1}=\tau^{\bar{1}}$. The forms $\theta_{1}{}^{1},\tau^{1}$ are called the pseudohermitian connection form and torsion form, respectively. Recall that the Heisenberg group $H_{1}$ is the space $\mathbb{R}^{3}$ endowed with the group multiplication

[TABLE]

which is a $3$-dimensional Lie group. The space of all left invariant vector fields is spanned by the following three vector fields:

[TABLE]

The standard contact bundle on $H_{1}$ is the subbundle $\mathring{\xi}$ of the tangent bundle $TH_{1}$, which is spanned by $\mathring{e}_{1}$ and $\mathring{e}_{2}$. It can also be equivalently defined as the kernel of the contact form

[TABLE]

The CR structure on $H_{1}$ is the endomorphism $\mathring{J}:\mathring{\xi}\rightarrow\mathring{\xi}$ defined by

[TABLE]

One can view $H_{1}$ as a pseudohermitian manifold with the standard pseudohermitian structure $(\mathring{J},\Theta)$. In the Heisenberg group $H_{1}$, relative to the standard left invariant frame $\mathring{Z}_{1}=\frac{1}{2}(\mathring{e}_{1}-i\mathring{e}_{2})$ (dual coframe is $\mathring{\theta}^{1}=dx+idy$), it is easy to see that both forms $\theta_{1}{}^{1}$ and $\tau^{1}$ vanish.

2.1. The deformation tensor

Suppose $J$ is a CR structure compatible with $\Theta$ in the following sense: it is defined on $\mathring{\xi}$ such that

[TABLE]

Let $Z_{1}$ be a CR holomorphic vector field relative to $J$. We express it as

[TABLE]

for some function $a_{1}{}^{1},b_{1}{}^{\bar{1}}$. We compute

[TABLE]

The compatibility of $J$ with $\Theta$ in (2.2) implies that $d\Theta(Z_{1},JZ_{1})>0$, which together with (2.4) implies that

[TABLE]

In particular, we have $a_{1}{}^{1}\neq 0$. Thus we can define

[TABLE]

where $a_{\bar{1}}{}^{\bar{1}},b_{\bar{1}}{}^{1}$ is the conjugate of $a_{1}{}^{1},b_{1}{}^{\bar{1}}$, respectively. We call $\phi$ the deformation tensor of $J$ (note that $\phi$ depends on frames. It behaves as a tensor when changing frames. For notational simplicity, we suppress its tensor indices). Thus (2.5) implies that $|\phi|<1$. It follows from (2.3) and (2.6) that any CR anti-holomorphic vector field $Z_{\bar{1}}$ has the form $Z_{\bar{1}}=a_{\bar{1}}{}^{\bar{1}}(\mathring{Z}_{\bar{1}}+\phi\mathring{Z}_{1})$, for some function $a_{\bar{1}}{}^{\bar{1}}$. Conversely, any function $\phi$ with $|\phi|<1$ defines a CR structure $J$ compatible with $\Theta$ by regarding $Z_{\bar{1}}=a_{\bar{1}}{}^{\bar{1}}(\mathring{Z}_{\bar{1}}+\phi\mathring{Z}_{1})$ as its corresponding CR anti-holomorphic vector field.

3. Proof

Let $(M^{3},J,\theta)$ be a pseudohermitian manifold. To prove Theorem A, first we would like to construct a sequence of pseudohermitian structures $\{(J_{i},\theta_{i})\}$ such that $\{(J_{i},\theta_{i})\}$ converges to $(J,\theta)$ in $C^{0}$ and the corresponding Tanaka-Webster scalar curvature $R_{i}$ also converges to the Tanaka-Webster scalar curvature $R$ of $(J,\theta)$ in $C^{0}$. This is Proposition 3.6. In addition, each CR structure $J_{i}$ we construct in Proposition 3.6 is CR spherical around a given point $p\in M$. Then, together with Proposition 3.7 and Theorem 1.1, we obtain Theorem A.

For Proposition 3.6, we construct such a sequence as follows: For each $p\in M$, there exists a neighborhood $U$ of $p\in M$ which is contactomorphic to a neighborhood $V$ of $0\in H_{1}$. Let $\Phi:U\rightarrow V$ be such a contactomorphism and $\Phi(p)=0$, we identify $U$ with $V$ under $\Phi$. Then, on $U$ (or on $V$), the CR structure $J$ can be represented by a deformation tensor $\phi$ with $|\phi|<1$ such that $\mathring{Z}_{\bar{1}}+\phi\mathring{Z}_{1}$ is a CR anti-holomorphic vector field. In addition, it is easy to see that one can take a contactomorphism $\Phi$ with $\Phi(p)=0$ such that the deformation function $\phi$ satisfies $\phi(0)=0\ \text{and}\ \phi_{1}(0)=\phi_{\bar{1}}(0)=0$, where $\phi_{1}=\mathring{Z}_{1}\phi$ and $\phi_{\bar{1}}=\mathring{Z}_{\bar{1}}\phi$. Thus, we can assume, without loss of generality, that

[TABLE]

Relative to the contact form $\Theta$,

[TABLE]

is a unit vector field. By (3.2), the dual coframe is

[TABLE]

On $U$, we can express the pseudohermitian connection form, torsion form, Tanaka-Webster curvature and sub-laplacian of $(J,\Theta)$ in terms of objects of the Heisenberg group as Propositions 3.1 and 3.2 specify. These expressions help us construct a pseudohermitian sequence we want.

Proposition 3.1**.**

Let $\theta_{1}{}^{1}$ and $\tau^{1}=A^{1}{}_{\bar{1}}\theta^{\bar{1}}$ be the pseudohermitian connection form and torsion form relative to $\theta^{1}$, respectively. Then we have

[TABLE]

where all the derivatives are computed in $H_{1}$; for example, $\phi_{1}=\mathring{Z}_{1}\phi,\ \phi_{0}=T\phi$, and so on.

Proof.

One can check directly that $\theta_{1}{}^{1}$ and $A^{1}{}_{\bar{1}}$ in (3.4) satisfy the structure equations (2.1). And by uniqueness, we complete the proof. ∎

Proposition 3.2**.**

Let $R^{\phi,\Theta}$ and $\Delta_{b}^{\phi,\Theta}$ be the Webster curvature and (negative) sub-laplacian of $(J,\Theta)$ on $U$, respectively. Then we have

[TABLE]

and

[TABLE]

Proof.

Recall that S. Webster [10] showed that $d\theta_{1}{}^{1}$ can be written as

[TABLE]

where $R$ is the Tanaka-Webster scalar curvature. Since $\theta^{1}$ is an unit coframe, we have $\theta^{1}\wedge\theta^{\bar{1}}=-id\Theta=\mathring{\theta}^{1}\wedge\mathring{\theta}^{\bar{1}}$. On the other hand, $d\mathring{\theta}^{1}=0$. Hence, (3.5) follows immediately from (3.4). For (3.6), recall that

[TABLE]

and (3.6) is just a straightforward computation in terms of (3.4). ∎

Remark 3.3**.**

The first author and I. H. Tsai deduced a more general formula for the Tanaka-Webster scalar curvature (see (4.6) in [2]).

To construct the sequence we want, we also need the following lemma which is a standard result in the literature (see [3, 7]).

Lemma 3.4**.**

*For any $\delta>0$, there is a nonnegative function $\chi_{\delta}\in C^{\infty}(\mathbb{R})$ such that

(i) $0\leq\chi_{\delta}\leq 1,\ \chi_{\delta}(t)\equiv 1$ in a neighborhood of [math] and $\chi_{\delta}(t)\equiv 0$ for $|t|\geq\delta$;

(ii) $|\chi_{\delta}^{\prime}(t)|\leq\delta t^{-1}$ and $|\chi_{\delta}^{\prime\prime}(t)|\leq\delta t^{-2}$ for all $t$.*

Now, for each $\delta>0$, by means of the cut-off function $\chi_{\delta}$, we define the CR structure $J^{\delta}$ with the corresponding deformation tensor $\phi^{\delta}=(1-\chi_{\delta}(\rho))\phi$. It is easy to see that $J^{\delta}=J$ outside the $\delta$-ball $B(\delta)$ centered at [math] and $J^{\delta}=\mathring{J}$ in a neighborhood of [math], which is CR spherical. If we, in addition, consider $\theta^{\delta}=\Theta$, then it is easy to see that the sequence $(J^{\delta},\theta^{\delta})$ converges to $(J,\theta)$ in $C^{0}$ (Note that we have chosen $\theta$ such that $\theta|_{U}=\Theta$). However, in general, since $R^{\phi,\Theta}(0)$ may not be zero, the corresponding Tanaka-Webster curvature of $(J^{\delta},\theta^{\delta})$ does not converge to the one of $(J,\theta)$. In order to have a sequence we want in Proposition 3.6, we need to deform the contact form $\Theta$.

Recall that if we consider the new contact form $\theta^{u}=u^{2}\Theta$, then, on $U$, we have the following transformation law of the Tanaka-Webster scalar curvature: (for the details, see [4, 6])

[TABLE]

where $R^{\phi,\theta^{u}}$ is the Tanaka-Webster scalar curvature with respect to $(J,\theta^{u})=(J,u^{2}\Theta)$.

On the other hand, the standard CR structure of the Heisenberg group on $U$ is represented by the zero deformation function $\phi\equiv 0$, which is CR spherical. Let $u$ be a positive function in a neighborhood of [math] such that $u(0)=1,(\mathring{Z}_{1}u)(0)=(\mathring{Z}_{\bar{1}}u)(0)=0$ and

[TABLE]

It follows from (3.9) and (3.10) that

[TABLE]

3.1. Construction of a sequence $(J^{\delta},\theta^{\delta})$

We are now ready to construct a sequence of pseudohermitian structures we describe in the beginning of this section. First, we re-formulate (3.5) and (3.6) as what we need.

Proposition 3.5**.**

Let $F=F(\phi)=\left(\frac{1}{1-|\phi|^{2}}\right)^{\frac{1}{2}}$. We have

[TABLE]

where $P_{ab},Q_{ab},P,S_{ab}$ and $S$ are all polynomials in $F,\phi,\bar{\phi},\phi_{1},\phi_{\bar{1}},\bar{\phi}_{1},\bar{\phi}_{\bar{1}}$ such that

[TABLE]

Since $F(0)=1$, condition (3.13) means that each polynomial does not include monomial terms $F^{k}$ for some nonnegative integer $k$.

Now, for each $\delta>0$, we define a pseudohermitian structure $(J^{\delta},\theta^{\delta})$ by

[TABLE]

Here, $\chi_{\delta}$ is the function constructed in Lemma 3.4. It follows from Lemma 3.4 that $(\phi^{\delta},\theta^{\delta})=(\phi,\Theta)$ outside the $\delta$-ball $B(\delta)$ centered at [math], and $(\phi^{\delta},\theta^{\delta})=(0,\theta^{u})$ in a neighborhood of [math]. Moreover, by (3.1), we have

[TABLE]

Notice that we have used the cut-off function $\chi_{\delta}$ to take the average of the two structures $(J,\theta)$ and $(\mathring{J},\theta^{u})$ on $U$, instead of the standard structure $(\mathring{J},\Theta)$, so that we have the first equation of (3.15) which will make sure later that we have Proposition 3.6.

3.2. Some uniform bounds

In this subsection, we provide some uniform bounds of invariant functions, which will be used in the proof of Proposition 3.6. Define

[TABLE]

Since $|\phi^{\delta}|\leq|\phi|$ by (3.14), we have $1\leq|F^{\delta}|\leq|F|$, and hence $|F^{\delta}|$ has an uniform bound. By (3.14), we have $(v^{\delta})^{2}-u^{2}=(1-\chi_{\delta})(1-u^{2})$, which together with Lemma 3.4 implies

[TABLE]

This implies that $v^{\delta}$ has an uniform bound. It follows from the definition of $\phi^{\delta}$ in (3.14) that, for $a,b\in\{1,\bar{1}\}$, the derivatives of $\phi^{\delta}$ are given by

[TABLE]

By (3.15), (3.17) and Lemma 3.4, $|\phi_{ab}|$ has an uniform upper bound for each $a,b\in\{1,\bar{1}\}$. Also, for $a,b\in\{1,\bar{1}\}$, it follows from (3.14) that the derivatives of $v^{\delta}$ are given by

[TABLE]

For the same reason, (3.15), (3.18) together with Lemma 3.4 show that $|(v^{\delta})_{a}|,|(v^{\delta})_{ab}|$ has an uniform upper bound for each $a,b\in\{1,\bar{1}\}$.

3.3. Statement and Proof of Proposition 3.6.

The sequence we construct in (3.14) satisfies suitable properties which we summarize in Proposition 3.6.

Proposition 3.6**.**

The sequence $\{(\phi^{\delta},\theta^{\delta})\}$ converges to $(\phi,\Theta)$ in $C^{0}$. The corresponding Tanaka-Webster scalar curvature $R^{\phi^{\delta},\theta^{\delta}}$ also converges to $R^{\phi,\Theta}$ in $C^{0}$.

Proof.

From the construction of $(\phi^{\delta},\theta^{\delta})$ in (3.14), and noting that $\phi(0)=0$ and $u(0)=1$, one can show that $\{(\phi^{\delta},\theta^{\delta})\}$ converges to $(\phi,\Theta)$ in $C^{0}$. Therefore we only need to show that $R^{\phi^{\delta},\theta^{\delta}}$ converges to $R^{\phi,\Theta}$ in $C^{0}$.

Since $\theta^{\delta}=(v^{\delta})^{2}\Theta$ where $(v^{\delta})^{2}=1+\chi_{\delta}(u^{2}-1)$, it follows from the transformation law of the Tanaka-Webster scalar curvature (3.9) that

[TABLE]

From above, we know that $v^{\delta}$ has an uniform bound and all $|\phi_{ab}|,|(v^{\delta})_{a}|,|(v^{\delta})_{ab}|$ and $|F^{\delta}|$ has an uniform upper bound. Using (3.12) with $\phi,u$ replaced by $\phi^{\delta},v^{\delta}$ respectively, together with (3.15), (3.16) and Lemma 3.4, we have

[TABLE]

for some positive constant $C$. Combining (3.19) and (3.20), we complete the proof of the proposition. ∎

3.4. Proof of Theorem A

Now we are ready to prove Theorem A. It suffices to prove the following proposition.

Proposition 3.7**.**

Let $\lambda(M,J^{\delta})$ be the CR Yamabe constant with respect to $J^{\delta}$, we have

[TABLE]

Proof.

Recall that we have constructed a sequence $(J^{\delta},\theta^{\delta})$ which converges to $(J,\theta)$ in $C^{0}$. In addition, $(J^{\delta},\theta^{\delta})=(J,\theta)$ outside the ball $B(\delta)$, and

[TABLE]

Notice that we have chosen the contact form $\theta$ such that $\theta|_{B(\delta)}=\Theta$. Let $dV=\theta\wedge d\theta$ and $dV^{\delta}=\theta^{\delta}\wedge d\theta^{\delta}=(v^{\delta})^{4}dV$. And let $R=R^{J,\theta}=R^{\phi,\Theta}$ and $R^{\delta}=R^{J^{\delta},\theta^{\delta}}=R^{\phi^{\delta},\theta^{\delta}}$. Since $(J^{\delta},\theta^{\delta})\rightarrow(J,\theta)$ in $C^{0}$, it is easy to see that, for any $\varepsilon>0$ with $\varepsilon\ll 1$, if $\delta$ is small enough, then we have

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and hence $|R^{\delta}|$ has an uniform bound. Next we need the following:

Lemma 3.8**.**

Given $\varepsilon>0$, if $\delta$ is small enough, then we have

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Proof of Lemma 3.8:.

First, by (3.2), we have

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Therefore, whenever $u_{\bar{1}}\neq 0$, we have

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if $\delta$ is small enough (since $\phi(0)=0$). Similarly, we have

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We have thus completed the proof of Lemma 3.8. ∎

Now, for each $\delta>0$, there exists a function $u^{\delta}$ such that

[TABLE]

We are going to estimate

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By (3.22) and Hölder inequality, we have

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and, by (3.22) and Lemma 3.8, we have

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since $|\int_{M}R^{\delta}(u^{\delta})^{2}\ dV^{\delta}|$ has an uniform upper bound. Substituting (3.25) and (3.26) into (3.24), we obtain

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Similarly, we have

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where $u^{0}$ is a function such that

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Since $\int_{M}(u^{0})^{4}dV=1$, we have

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by (3.22). Similarly, we have

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By (3.27)-(3.30), we can get

[TABLE]

This completes the proof of Proposition 3.7. ∎

Therefore, if $\delta$ is small enough, we have $\lambda(M,J^{\delta})>0$ by Proposition 3.7 and the assumption that $\lambda(M,J)>0$. By construction, each $J^{\delta}$ is spherical around the point $p\in M$. To complete the proof of Theorem A, we choose $\delta_{1},\delta_{2}$ so that the CR Yamabe constants $\lambda(M_{1},J_{1}^{\delta_{1}})$ and $\lambda(M_{2},J_{2}^{\delta_{2}})$ are both positive. Then, by using the argument in [1] (see the paragraph after Theorem 1.1) to glue $M_{1}$ and $M_{2}$ by a Heisenberg cylinder, we get a CR structure on the connected sum $M_{1}\#M_{2}$ with positive CR Yamabe constant.