Flexible and inflexible $CR$ submanifolds

TL;DR

This paper establishes new embedding results for compactly supported deformations of certain $CR$ submanifolds in complex Euclidean space, showing stability of embeddability under specific pseudoconcavity conditions and providing examples of non-embeddable deformations.

Contribution

It proves that 2-pseudoconcave $CR$ submanifolds retain embeddability under compact deformations, extending previous quadratic case results and providing counterexamples.

Findings

Stable embeddability for 2-pseudoconcave $CR$ submanifolds

Extension of previous quadratic $CR$ submanifold results

Existence of weakly $2$-pseudoconcave $CR$ manifolds with non-embeddable deformations

Abstract

In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of $\mathbb{C}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in $\mathbb{C}^{n+d}$ manifolds. This improves an earlier result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$ submanifold of $\mathbb{C}^{n+d}$. We also give examples of weakly $2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$ deformations that are not even locally $CR$ embeddable.

Equations

$$[T^{0,1}M,T^{0,1}M]\subset T^{0,1}M$$

$$T^{0,1}M=\{X+iJX\mid X\in \Gamma (M,HM)\}\subset \Gamma (M,\mathbb{C}TM),$$

$${\mathcal{L}}_p(\xi ,X)=\xi ([J\tilde{X},\tilde{X}])=d\tilde{\xi }(X,JX)\mathrm{for}X\in H_{p}M$$

$$\Vert u{\Vert }^2\le C(\Vert {\overline{\partial }}_{M}u{\Vert }^2+\Vert {\overline{\partial }}_M^{\ast }u{\Vert }^2)$$

$$\Vert u{\Vert }_{\epsilon }^2\le C_K(\Vert {\overline{\partial }}_{M_a}u{\Vert }^2+\Vert {\overline{\partial }}_{M_a}^{\ast }u{\Vert }^2+\Vert u{\Vert }^2)$$

$$\Vert u{\Vert }^2\le C(\Vert {\overline{\partial }}_{M_a}u{\Vert }^2+\Vert {\overline{\partial }}_{M_a}^{\ast }u{\Vert }^2)$$

$$\Vert u_{a_{\nu }}\Vert =1,$$

$$\Vert {\overline{\partial }}_{M_{a_{\nu }}}{u}_{a_{\nu }}{\Vert }^2+\Vert {\overline{\partial }}_{M_{a_{\nu }}}^{\ast }{u}_{a_{\nu }}{\Vert }^2<a_{\nu }.$$

$$\Vert \chi u{\Vert }^2\le C(\Vert {\overline{\partial }}_M(\chi u){\Vert }^2+\Vert {\overline{\partial }}_M^{\ast }(\chi u){\Vert }^2)$$

$$\Vert \chi u{\Vert }^2\le C^{\prime }(\Vert {\overline{\partial }}_{M}u{\Vert }^2+\Vert {\overline{\partial }}_M^{\ast }u{\Vert }^2+{\int }_{K_1\setminus K}|u{|}^{2}dV)$$

$$\begin{array}[]{cccc}T_{f}:&L^{2}_{n+d,n}(M_{a}\cap B)&\longrightarrow&\mathbb{C}\\ &\psi&\mapsto&\int_{M_{a}\cap B}f\wedge\varphi,\end{array}$$

$${\int }_{M_a\cap B}f\wedge \phi ={\int }_{M_a\cap B}f\wedge {\overline{\partial }}_{M_a}h={\int }_{M_a\cap B}{\overline{\partial }}_{M_a}(f\wedge h)=0.$$

$${\int }_{M_a\cap B}u\wedge {\overline{\partial }}_{M_a}\phi =T_f({\overline{\partial }}_{M_a}\phi )={\int }_{M_a\cap B}f\wedge \phi$$

$$\Vert \zeta u{\Vert }_{m+\epsilon }^2\le C_{K,m}(\Vert {\zeta }_1{\overline{\partial }}_{M_a}u{\Vert }_m^2+\Vert {\zeta }_{1}u{\Vert }^2)$$

$$\{\begin{array}{rlr}{\overline{X}}_0& =& \frac{\partial }{\partial {\overline{z}}_{\beta }}\\ {\overline{X}}_j& =& {\overline{L}}_j+\beta a\omega ({\overline{L}}_j)z_{\beta }\frac{\partial }{\partial z_{\beta }},j=1,2\end{array}$$

$$0={\overline{\partial }}_M({\overline{L}}_1,{\overline{L}}_2)={\overline{L}}_1(\omega ({\overline{L}}_2))-{\overline{L}}_2(\omega ({\overline{L}}_1))-\omega ([{\overline{L}}_1,{\overline{L}}_2]),$$

$$[{\overline{X}}_1,{\overline{X}}_2]$$

$$\phi :(t,z_{\beta })\mapsto (z_1(t,z_{\beta }),\dots ,z_{2+d}(t,z_{\beta }),u_{3+d}(t,z_{\beta }))$$

$$F(w)=\varphi_{\ast}\big{(}-[\frac{\partial u_{3+d}}{\partial z_{\beta}}]^{-1}\big{)},$$

$$G(w)={\int }_0^{w_{3+d}}F(w_1,\dots ,w_{2+d},\eta )d\eta$$

$$\begin{gathered} \frac{\partial g}{\partial z_{\beta }}=F(w_1,\dots ,w_{2+d},u_{3+d}(t,z_{\beta }))\frac{\partial u_{3+d}}{\partial z_{\beta }}(t,z_{\beta }) \\ =-1, \end{gathered}$$

Full text

1 Introduction

In a previous paper [BH2] we introduced the concept of flexible versus inflexible $CR$ submanifolds. This is related to the $CR$ embeddability of deformations of $CR$ structures. Roughly speaking a flexible submanifold admits a compactly supported $CR$ deformation that ”pops out” of the space of globally $CR$ embeddable manifolds. On the other hand, for an inflexible $CR$ submanifold, any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable manifolds.

Much work has been concentrated on $CR$ manifolds $M$ of hypersurface type which form the boundaries of strictly pseudoconvex domains. In that situation, $M$ is inflexible when $\dim_{CR}M\geq 2$, and $M$ is flexible when $\dim_{CR}M=1$ (even without the assumption of strict pseudoconvexity). See example 1 in section 4.

Even in the situation of $\mathrm{codim}_{CR}M=1$ (hypersurface type) it is of interest to study what happens for split signature of the Levi form. In that hypersurface case, 1-pseudoconcavity means that the Levi form has at least 1 negative eigenvalue, and at least 1 positive eigenvalue; 2-pseudoconcavity means that the Levi form has at least 2 negative and at least 2 positive eigenvalues, etc.

And $CR$ manifolds can have higher $CR$-codimension, in which case $q$-pseudoconcavity also seems to be a fruitful concept. It means that for every $x\in M$ and every characteristic conormal direction $\xi$ at $x$, the scalar Levi form $\mathcal{L}_{x}(\xi,\cdot)$ in this conormal direction has at least $q$ positive and $q$ negative eigenvalues. (See section 2 for the precise definitions.)

The theory of pseudoconcave $CR$ manifolds was initiated approximately 25 years ago (see [HN]). Since that time it has slowly come to light that $CR$ manifolds of higher codimension arise naturally in mathematics; i.e., such manifolds abound, but for a long time it was ignored that they have a natural $CR$ structure. Besides typical examples of quadratic $CR$ submanifolds of $\mathbb{C}^{n+d}$, they also arise naturally as minimal orbits for the holomorphic action of real Lie groups on flag manifolds. These are even homogeneous and almost always are $q$-pseudoconcave, for some $q$. In fact in [MN] the authors follow the general method initiated by N. Tanaka of investigating manifolds endowed with partial complex structures that come from Levi-Tanaka algebras which are the canonical prolongations of pseudocomplex fundamental graded Lie algebras. A lot of explicit such examples can be found in [MN], [HN] or [HN1].

When $M$ is of hypersurface type, there are some hints that the 1-pseudoconcave case (Lorentzian case) and the $q$-pseudoconcave ($q\geq 2$) differ. For example, it is in the Lorentzian signature case where it is possible to generalize Nirenberg’s example [Ni] to $\dim_{CR}>1$, as was done in [JT]. But when $q\geq 2$, that construction does not work. Indeed in example 6 of section 4 we present a $CR$ manifold $N$, of any CR-codimension, which is only 1-pseudoconcave (but weakly 2-pseudoconcave) and it is flexible. This shows that our Theorem 1.1 below is almost optimal. However, our $N$ is not globally $CR$ embedded into Euclidean space. Therefore it remains an open problem to find a $1$-pseudoconcave $CR$ submanifold of some Euclidean space that is flexible.

The main result obtained in [BH2] was that any $2$-pseudoconcave quadratic $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$ is inflexible. In the present paper we obtain the same result for $CR$ submanifolds that are not necessarily assumed to be quadratic. More precisely,

Theorem 1.1

*Let $M$ be a $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$ that is $2$-pseudoconcave. Let $(M_{a},HM_{a},J_{a})_{|a|<a_{o}}$ be a compactly supported $CR$ deformation of $(M,HM,J)$. Then, provided $a$ is sufficiently small, given any smooth $CR$ function $f:(M,HM,J)\longrightarrow\mathbb{C}$, there is a $CR$ function $f_{a}:(M_{a},HM_{a},J_{a})\longrightarrow\mathbb{C}$ such that for any given $\ell\in\mathbb{N}$, any given compact $K$ of $M$ and arbitrary small $\varepsilon>0$, one can find a $CR$ function $f_{a}:(M_{a},HM_{a},J_{a})\longrightarrow\mathbb{C}$ such that the $\mathcal{C}^{\ell}$ norm of $f-f_{a}$ on $K$ is less than $\varepsilon$.

Moreover, $f_{a}$ can be chosen to coincide with the given $f$ outside a compact of $M$. In particular, $(M_{a},HM_{a},J_{a})$ is $CR$ embeddable into $\mathbb{C}^{n+d}$ for $a$ sufficiently close to [math].*

Corollary 1.2

Let $M$ be a 2-pseudoconcave $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$. Then $M$ is inflexible.

Remark. The same result holds, with the same proof, if $\mathbb{C}^{n+d}$ is replaced by a strictly pseudoconvex domain in $\mathbb{C}^{n+d}$. We conjecture that it also holds with $\mathbb{C}^{n+d}$ is replaced by an $(n+d)$-dimensional Stein manifold $X$. However, our proof relies on the results from [LS]; and it is not clear if these results also hold in the more general setting of Stein manifolds.

In the proof of Theorem 1.1 we use an $L^{2}$ vanishing result obtained in [LS], which involved heavy use of integral formulas. In [BH2] we were able to obtain the analogous result by employing partial Fourier transform techniques, because of the quadratic nature of $M$. However in both [BH2] and in the present paper we also need certain subelliptic estimates, from [FK] in codimension one, and from [HN] in higher codimension. Thus, although the results of [BH2] were restricted to the quadratic case, the proofs there are more self-contained, since they do not rely on the rather complicated integral formulas upon which [LS] is based.

2 Definitions

An abstract $CR$ manifold of type $(n,d)$ is a triple $(M,HM,J)$, where $M$ is a smooth real manifold of dimension $2n+d$, $HM$ is a subbundle of rank $2n$ of the tangent bundle $TM$, and $J:HM\rightarrow HM$ is a smooth fiber preserving bundle isomorphism with $J^{2}=-\mathrm{Id}$. We also require that $J$ be formally integrable; i.e. that we have

[TABLE]

where

[TABLE]

with $\Gamma$ denoting smooth sections.

The $CR$ dimension of $M$ is $n\geq 1$ and the $CR$ codimension is $d\geq 1$.

$M$ admits a $CR$ embedding into some complex manifold $X$ if one can find a smooth embedding $\varphi$ of $M$ into $X$ such that the induced $CR$ structure $\varphi_{\ast}(T^{0,1}M)$ on $\varphi(M)$ coincides with the $CR$ structure $T^{0,1}(X)\cap\mathbb{C}T(\varphi(M))$ from the ambient complex manifold $X$.

Let $(M,HM,J)$ be a $CR$ manifold of type $(n,d)$ globally $CR$ embedded into some complex manifold $X$. We say that $(M,HM,J)$ admits a compactly supported $CR$ deformation if there exists a family $(M_{a},HM_{a},J_{a})_{|a|<a_{o}}$ of abstract $CR$ manifolds depending smoothly on a real parameter $a$, $|a|<a_{o}$ and converging to $(M,HM,J)$ as $a$ tends to [math] in the usual $\mathcal{C}^{\infty}$ topology; we also require that $(M_{a},HM_{a},J_{a})=(M,HM,J)$ for every $a\not=0$ outside some compact $K$ of $M$ not depending on $a$.

Note that when $(M,HM,J)$ is $CR$ embedded into some complex manifold, then one can always ”punch” $M$ as to obtain compactly supported $CR$ deformations (at least locally). With the exception of $n=1$ (when the formal integrability condition is always satisfied), it can be difficult, however, to find compactly supported $CR$ deformations in the absence of local $CR$ embeddability.

We say that $(M,HM,J)$ is a flexible $CR$ submanifold of $X$ if it admits a compactly supported $CR$ deformation $(M_{a},HM_{a},J_{a})_{|a|<a_{o}}$ such that for every sufficiently small $a\not=0$, the $CR$ structure $(M_{a},HM_{a},J_{a})$ is not globally $CR$ embeddable into $X$. So, for example, the Heisenberg $CR$ structure $\mathbb{H}^{2}$ in $\mathbb{C}^{2}$ is flexible. This follows from Nirenberg’s famous local nonembeddability examples [Ni], which can be interpreted as small (local) deformations of the Heisenberg structure on $\mathbb{H}^{2}$. More examples will be discussed in the last section.

We say that $(M,HM,J)$ is an inflexible $CR$ submanifold of $X$ if it is not flexible. That means that $(M,HM,J)$ is inflexible if and only if for every compactly supported $CR$ deformation $(M_{a},HM_{a},J_{a})_{|a|<a_{o}}$ of $(M,HM,J)$, the $CR$ manifold $(M_{a},HM_{a},J_{a})$ is globally $CR$ embeddable into $X$.

We denote by $H^{o}M=\{\xi\in T^{\ast}M\mid<X,\xi>=0,\forall X\in H_{\pi(\xi)}M\}$ the characteristic conormal bundle of $M$. Here $\pi:TM\longrightarrow M$ is the natural projection. To each $\xi\in H^{o}_{p}M\setminus\{0\}$, we associate the Levi form at $p$ in the codirection $\xi:$

[TABLE]

which is Hermitian for the complex structure of $H_{p}M$ defined by $J$. Here $\tilde{\xi}$ is a section of $H^{o}M$ extending $\xi$ and $\tilde{X}$ a section of $HM$ extending $X$.

Following [HN] $M$ is called $q$-pseudoconcave, with $0\leq q\leq\frac{n}{2}$, if for every $p\in M$ and every characteristic conormal direction $\xi\in H^{o}_{p}M\setminus\{0\}$, the Levi form $\mathcal{L}_{p}(\xi,\cdot)$ has at least $q$ negative and $q$ positive eigenvalues.

For other standard definitions related to $CR$ structures we also refer the reader to [HN] or [HN1].

3 Proofs

The idea of the proof of Theorem 1.1 is as follows: For a given $CR$ function $f$ on $M$ we want to find a $CR$ function $f_{a}$ on $M_{a}$ which is very close to the given $f$ on $M$. Therefore we want to solve the Cauchy-Riemann equations $\overline{\partial}_{M_{a}}u=\overline{\partial}_{M_{a}}f$ with $u$ having compact support and the $\mathcal{C}^{k}$-norms of $u$ being controlled by some $\mathcal{C}^{l}$-norms of $\overline{\partial}_{M_{a}}$ (uniformly with respect to $a$). Setting $f_{a}=f-u_{a}$ then gives the desired $CR$ function on $M_{a}$.

Let $M$ be as in Theorem 1.1, and let $B$ be a sufficiently large Euclidean ball containing the compact $K$ that is the support of the $CR$ deformation of $M\subset\mathbb{C}^{n+d}$. Recalling that $M$ is $2$-pseudoconcave, we have the following result from [LS, Theorem 1.0.2]:

Proposition 3.1

Let $q=n-1$ or $q=n$, and assume $f\in L^{2}_{n+d,q}(M\cap B)$ satisfies $\overline{\partial}_{M}f=0$. Then there exists $u\in L^{2}_{n+d,q-1}(M\cap B)$ satisfying $\overline{\partial}_{M}u=f$.

Here we are considering (unweighted) $L^{2}$ spaces with respect to the induced metrics from the Euclidean metric on $\mathbb{C}^{n+d}$. By classical Hilbert space theory (see e.g. [H, Theorem 1.1.2]), one deduces from Proposition 3.1 the following

Proposition 3.2

Let $q=n-1$ or $q=n$. Then there exists a constant $C>0$ such that

[TABLE]

for all $u\in L^{2}_{n+d,q}(M\cap B)\cap\mathrm{Dom}(\overline{\partial}_{M})\cap\mathrm{Dom}(\overline{\partial}_{M}^{\ast})$.

Next, we use again that $M$ is 2-pseudoconcave. 2-pseudoconcavity is clearly stable under smooth, small perturbations. Therefore $M_{a}$ is also 2-pseudoconcave for $a$ sufficiently small, and the 2 positive resp. 2 negative eigenvalues of the Levi form in sufficiently close characteristic conormal directions can be bounded from below resp. above independent of $a$. Therefore one obtains a uniform subelliptic estimate in degrees $q\in\{0,1,n-1,n\}$ (by closely looking at the proofs in [FK] for $d=1$ and [HN] for higher codimensions): There exists $\varepsilon>0$ such that for every compact $K$ of $M$, there exists a constant $C_{K}>0$ independent of $a$ such that

[TABLE]

for all smooth forms $u\in\mathcal{D}_{K}^{p,q}(M_{a})$ with support contained in $K$, $0\leq p\leq n+d$, $q\in\{0,1,n-1,n\}$.

Combining Proposition 3.2 and (3.1), we can establish an $L^{2}$ a priori estimate in degree $(n+d,n-1)$ and $(n+d,n)$, which is uniform with respect to $a$ (in the sense that the constant involved does not depend on $a$).

Proposition 3.3

There is $a_{0}>0$ and a constant $C>0$ such that for $q\in\{n-1,n\}$ we have

[TABLE]

for all $u\in L^{2}_{n+d,q}(M\cap B)\cap\mathrm{Dom}(\overline{\partial}_{M_{a}})\cap\mathrm{Dom}(\overline{\partial}_{M_{a}}^{\ast})$, $|a|<a_{0}$.

Proof. Assume by contradiction that there is a sequence $\{u_{a_{\nu}}\}\in L^{2}_{n+d,q}(M_{a_{\nu}}\cap B)\cap\mathrm{Dom}(\overline{\partial}_{M_{a_{\nu}}})\cap\mathrm{Dom}(\overline{\partial}^{\ast}_{M_{a_{\nu}}})$, $a_{\nu}\rightarrow 0$, such that

[TABLE]

whereas

[TABLE]

We now want to show that $\{u_{a_{\nu}}\}$ is a Cauchy sequence.

Remember that $M_{a_{\nu}}=M$ outside $K$. We now choose a slightly larger compact $K_{1}$ containing $K$ in its interior, and a smooth cut-off function $\chi$ such that $\chi\equiv 1$ outside $K_{1}$ and $\chi\equiv 0$ in a neighborhood of $K$. Since $\overline{\partial}_{M_{a_{\nu}}}$, $\overline{\partial}^{\ast}_{M_{a_{\nu}}}$ coincide with $\overline{\partial}_{M}$, $\overline{\partial}^{\ast}_{M}$ outside $K$, we obtain from Proposition 3.2

[TABLE]

for all $u\in L^{2}_{n+d,q}(M_{a}\cap B)\cap\mathrm{Dom}(\overline{\partial}_{M_{a_{\nu}}})\cap\mathrm{Dom}(\overline{\partial}^{\ast}_{M_{a_{\nu}}})$, which implies

[TABLE]

for some constant $C^{\prime}>0$.

On the other hand, let $\eta$ be a smooth cut-off function so that $\eta\equiv 1$ in a neighborhood of $K_{1}$. Then $\|\eta u_{a_{\nu}}\|_{\varepsilon}$ is bounded by (3.1), so the generalized Rellich lemma implies that the sequence $\{u_{a_{\nu}}\}$ restricted to $K_{1}$ is precompact in $L^{2}_{n+d,q}(K_{1})$. Thus it is no loss of generality to asume that the restriction of $\{u_{a_{\nu}}\}$ to $K_{1}$ is a Cauchy sequence. But this combined with (3.4) implies that $\{u_{a_{\nu}}\}$ is a Cauchy sequence in $L^{2}_{n+d,q}(M\cap B)$.

Denote by $u_{0}$ the limit of this sequence. From (3.3) it follows that $\overline{\partial}_{M}u_{0}$ and $\overline{\partial}^{\ast}_{M}u_{0}$, defined in the distribution sense, both vanish. But from (3.2) it also follows that $\|u_{0}\|=1$. This contradicts Proposition 3.2 and therefore completes the proof of the proposition. $\square$

By duality, we obtain from Proposition 3.3 that one can solve the $\overline{\partial}_{M_{a}}$-equation with support in $M\cap\overline{B}$ in degree $(0,1)$ with a uniform constant. For this, we consider an $L^{2}$ variant of $\overline{\partial}_{M_{a}}$ defined in the following way: Let $u\in L^{2}_{p,q}(M_{a}\cap B)$. We say that $u\in\mathrm{Dom}(\overline{\partial}^{c}_{M_{a}})$ and $\overline{\partial}_{M_{a}}^{c}u=f$ if there exists a sequence of test forms $u_{j}\in\mathcal{D}^{p,q}(M_{a}\cap B)$ such that $u_{j}\rightarrow u$ in $L^{2}$ and $\overline{\partial}_{M_{a}}u_{j}\rightarrow f$ in $L^{2}$.

Proposition 3.4

There is $a_{0}>0$ and a constant $C>0$ independent of $a$ such that for every $f\in L^{2}_{0,1}(M_{a})$ with $\overline{\partial}_{M_{a}}f=0$ and $f$ compactly supported in $M\cap B$, one can find $u\in L^{2}_{0,0}(M_{a})$ such that $\overline{\partial}_{M_{a}}^{c}u=f$ and $\|u\|\leq C\|f\|$.

Proof. Consider the operator

[TABLE]

where $\varphi\in L^{2}_{n+d,n-1}(M_{a}\cap B)$ satisfies $\overline{\partial}_{M_{a}}\varphi=\psi$ in the weak sense and $\|\varphi\|\leq C\|\psi\|$ (such a $\varphi$ exists by Proposition 3.3). $T_{f}$ is well defined. Indeed, if $\overline{\partial}_{M_{a}}\varphi=0$, then we may apply Proposition 3.3 again and conclude that there exists $h\in L^{2}_{n+d,n-2}(M_{a}\cap B)$ satisfying $\overline{\partial}_{M_{a}}h=\varphi$. By Stokes’ theorem this implies

[TABLE]

Note also that $T_{f}$ is continuous of norm $\leq C$. Using Riesz’ theorem, we conclude that there exists $u\in L^{2}_{0,0}(M_{a})$ satisfying

[TABLE]

for all $\varphi\in L^{2}_{n+d,n}(M_{a}\cap B)$. Let $\vartheta_{a}$ be the formal adjoint of $\overline{\partial}_{M_{a}}$ on $L^{2}_{\cdot,\cdot}(M_{a}\cap B)$. It is easy to see that $\overline{\partial}^{c}_{M_{a}}$ and $\vartheta_{a}$ are adjoint operators on $L^{2}_{\cdot,\cdot}(M_{a}\cap B)$. (3.5) implies that $(u,\vartheta_{a}\varphi)=(f,\varphi)$ for any $\varphi\in\mathrm{Dom}(\vartheta_{a})$, which is equivalent to $\overline{\partial}^{c}_{M_{a}}u=f$. $\square$

Proof of theorem 1.1.

Let $f$ be a $CR$ function on $M$. Then $\overline{\partial}_{M_{a}}f$ has compact support and tends to zero when $a$ tends to zero. Proposition 3.4 implies that we can solve the equation $\overline{\partial}_{M_{a}}u_{a}=\overline{\partial}_{M_{a}}f$ with $\|u_{a}\|\leq C\|\overline{\partial}_{M_{a}}f\|$ and $u_{a}$ supported in $M\cap\overline{B}$. Hence $u_{a}$ is as small as we wish in $L^{2}_{0,0}(M_{a})$, provided $a$ is small enough. It is well-known that the subelliptic estimate (3.1) in degree $q=0$ implies also the following: Suppose given a compact $K^{\prime}\subset M_{a}$ and two smooth real functions $\zeta,\ \zeta_{1}$ with $\mathrm{supp}\zeta\subset\mathrm{supp}\zeta_{1}\subset K^{\prime}$ and $\zeta_{1}=1$ on $\mathrm{supp}\zeta$, then for any integer $m\in\mathbb{N}$ there exists a constant $C_{K,m}$ such that

[TABLE]

Here $\|\ \|_{m}$ denotes the Sobolev norm of order $m$. But then also the $\mathcal{C}^{\ell}$-norm of $u_{a}$ over a given compact $K^{\prime}\subset M_{a}$ can be controlled by some $\mathcal{C}^{m}$-norm of $\overline{\partial}_{M_{a}}u_{a}=f$, and hence made small when letting $a$ tend to zero. Setting $f_{a}=f-u_{a}$ proves the theorem. $\square$

4 Examples of flexible $CR$ submanifolds

The aim of this section is to provide known and new examples of flexible $CR$ submanifolds.

Rossi [R] constructed small real analytic deformations of the standard $CR$ structure on the 3-sphere $S^{3}$ in $\mathbb{C}^{2}$, and such that the resulting abstract $CR$ structures fail to $CR$ embed globally into $\mathbb{C}^{2}$. Hence $S^{3}$ is a flexible $CR$ submanifold of $\mathbb{C}^{2}$.

This is in contrast to higher dimensions: Any strictly pseudoconvex $CR$ manifold $M$ of $CR$ dimension $n\geq 2$ is globally $CR$ embeddable into some $\mathbb{C}^{N}$ by [BdM]. If, in addition, $M$ is the boundary of a strictly pseudoconvex domain in $\mathbb{C}^{n+1}$, then $M$ is inflexible. This follows from a result by [T], since in this situation we have $H^{0,1}(M)=0$. 2. 2.

Nirenberg’s famous local nonembeddability examples [Ni] can be interpreted as small (local) deformations of the Heisenberg structure on $\mathbb{H}^{2}\subset\mathbb{C}^{2}$. Since the formal integrability condition for $CR$ structures is always satisfied in dimension $3$, one can use a cut-off function to make the local deformations a compactly supported deformation of the global Heisenberg group. 3. 3.

More generally, any $3$-dimensional $CR$ submanifold is flexible. Indeed, if $M$ has a point of strict pseudoconvexity, then one can use the local nonembeddability result of [JT] to produce a small, non-locally embeddable $CR$ deformation which is compactly supported near that point.

If $M$ is Levi-flat, then one first makes arbitrary small bumps near a fixed point to get points of strict pseudoconvexity and proceeds as before. 4. 4.

$S^{3}\times S^{3}\in\mathbb{C}^{4}$ is an example of a flexible $CR$ submanifold of codimension $2$ (because each factor is flexible). Depending on the conormal direction, its Levi-forms have signature $++$, $--$, $+0$ or $-0$. By adding more products one can obtain flexible $CR$ submanifolds of any $CR$ codimension. 5. 5.

Let $X$ be any compact Riemann surface. Then $S^{3}\times X$ is flexible. 6. 6.

Let $M$ be a compact $1$-pseudoconcave $CR$ submanifold of type $(2,d)$ of some complex manifold $X$, $d$ arbitrary. Then $N=M\times\mathbb{C}\mathbb{P}^{1}$ is again $1$-pseudoconcave, and even weakly $2$-pseudoconcave (the Levi form has signature $(+-0)$ in every nonzero conormal direction). Using ideas from [Hi1] we will now show that $N$ is flexible, which indicates that Theorem 1.1 is close to being optimal. Indeed, by Theorem 3.2 of [BH1] there exists a smooth $(0,1)$-form $\omega$ on $M$ satisfying $\overline{\partial}_{M}\omega=0$ on $M$ such that $\omega$ is not $\overline{\partial}_{M}$-exact on any neighborhood of any point $p$ on $M$. We will use this form $\omega$ to deform the $CR$ structure on $M\times\mathbb{C}\mathbb{P}^{1}$.

On $\mathbb{C}\mathbb{P}^{1}$ we use the two standard holomorphic charts $V_{+}=\mathbb{C}\mathbb{P}^{1}\setminus\{\infty\}$ and $V_{-}=\mathbb{C}\mathbb{P}^{1}\setminus\{\infty\}$ given by the stereographic projection, with coordinates $z_{+}\in V_{+}\simeq\mathbb{C}$ and $z_{-}\in V_{-}\simeq\mathbb{C}$, where $z_{-}z_{+}=1$ on $V_{-}\cap V_{+}$. Then the usual complex structure on $\mathbb{C}\mathbb{P}^{1}$ is given by $\frac{\partial}{\partial\overline{z}_{\beta}}$ on $V_{\beta}$ for $\beta=(+,-)$.

Let $U$ be an open set of $M$ such that $T^{0,1}M$ is spanned over $U$ by $\overline{L}_{1},\overline{L}_{2}$. We define $T^{0,1}N_{a}$ to be spanned over $U\times V_{\beta}$ by the basis

[TABLE]

This gives a well defined $CR$ structure on $N$. To see that the integrability condition is valid, first note that $[\overline{X}_{0},\overline{X}_{j}]=0$ for $j=1,2$. Moreover, by assumption on $\omega$ we have

[TABLE]

thus

[TABLE]

thus $T^{0,1}N_{a}$ is stable under the Lie bracket.

However, for $a\not=0$, local $CR$ embeddability of $N_{a}$ implies the local $\overline{\partial}_{M}$-exactness of $\omega$. The argument follows [Hi2] or [Hi3]. In fact the argument shows that $N_{a}$ is not even locally $CR$ embeddable at any point $(t^{o},z^{o}_{\beta})$ of $N_{a}=M\times\mathbb{C}\mathbb{P}^{1}$: Near $t^{o}$, $M$ is locally $CR$ embeddable into $\mathbb{C}^{2+d}$ with coordinate functions $\zeta_{1},\ldots,\zeta_{2+d}$. We may assume that $t=(t_{1},\ldots,t_{4+d})=(\mathrm{Re}\zeta_{1},\ldots,\mathrm{Re}\zeta_{2+d},\mathrm{Im}\zeta_{1},\mathrm{Im}\zeta_{2})$ are real coordinates on $M$ with $t^{o}=0$ in these coordinates.

Suppose now that we have a local $CR$ embedding of $N_{a}$ near $(t^{o},z^{o}_{\beta})$ by $CR$ functions $u_{1}(t,z_{\beta}),u_{2}(t,z_{\beta}),\ldots,u_{3+d}(t,z_{\beta})$ with $du_{1}\wedge\ldots\wedge du_{3+d}\not=0$ at $(t^{o},z^{o}_{\beta})$. Then each $u_{j}$ is holomorphic in $z_{\beta}$ in view of (4.1).

It is then not difficult to see that $\frac{\partial u_{j}}{\partial z_{\beta}}\not=0$ for some $j$ at $(t^{o},z^{o}_{\beta})$. By renaming, we may assume $\frac{\partial u_{3+d}}{\partial z_{\beta}}\not=0$.

The coordinates on $(z_{1},\ldots,z_{3+d})$ on $\mathbb{C}^{3+d}$ also define $CR$ functions on $N_{a}$ and $dz_{1}\wedge\ldots\wedge dz_{2+d}\wedge du_{3+d}\not=0$ at $(t^{o},u^{o}_{\beta})$. So we arrive at a new local embedding map

[TABLE]

of some neighborhood $W$ of $(t^{o},z_{\beta}^{o})$ into $\mathbb{C}^{3+d}$. $\varphi(W)$ is a piece of a real hypersurface in $\mathbb{C}^{3+d}$. Let $w=(w_{1},\ldots,w_{3+d})$ denote the coordinates in $\mathbb{C}^{3+d}$, and consider, for points on $\varphi(W)$, the function

[TABLE]

where $\varphi_{\ast}$ is the push-forward by the diffeomorphism $\varphi$ of $W$ onto $\varphi(W)$. It follows that $F$ is a $CR$ function on $\varphi(W)$; in particular, it is holomorphic in $w_{3+d}$ by the inverse mapping theorem for holomorphic functions of one variable. On $\varphi(W)$ we may define the function

[TABLE]

by a contour integral in the $w_{3+d}$-plane. This is well defined by the open mapping theorem from one complex variable. We now pull back to get a function $g(t,z_{\beta})=\varphi^{\ast}G$ on $V$, which is a $CR$ function there. This can be seen by replacing $F$ in (4.2) by a smooth extension $\tilde{F}$ of $F$ off of $\varphi(W)$ such that $\overline{\partial}\tilde{F}_{\mid\varphi(V)}=0$ and differentiating. Next we have

[TABLE]

so $g(t,z_{\beta})=-z_{\beta}+\chi(t)$, where $\chi(t)$ is a smooth ”constant of integration”. Now the fact that $g$ is a $CR$ function implies that $\overline{X}_{j}g=0$, hence $\overline{L}_{j}\chi-\beta a\omega(\overline{L}_{j})=0$ for $j=1,2$. But for $a\not=0$, this means that there exists a neighborhood $V^{\prime}$ of $t^{o}$ on $M$ such that $\omega$ is $\overline{\partial}_{M}$-exact on $V^{\prime}$. This is a contradiction to the assumption on $\omega$. Therefore for $a\not=0$, $N_{a}$ is not locally $CR$ embeddable on any open neighborhood of $(t^{o},z_{\beta}^{o})$ on $N_{a}$. $\square$