Boundary problem for Levi flat graphs

TL;DR

This paper proves that for a specific class of boundary graphs over convex domains in complex space, the Levi-flat hypersurface they bound is necessarily smooth and non-singular, extending previous general conditions.

Contribution

It establishes that when the boundary is a graph over a strongly convex domain, the Levi-flat hypersurface is also a smooth graph, ensuring non-singularity in this case.

Findings

Levi-flat hypersurface is a smooth graph over the domain

Hypersurface is necessarily non-singular

Extends previous conditions to specific boundary graphs

Abstract

In an earlier paper the authors provided general conditions on a real codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, such that there exists a possibly singular Levi-flat hypersurface $M$ bounded by $S$. In this paper we consider the case when $S$ is a graph of a smooth function over the boundary of a bounded strongly convex domain $\Omega\subset C^{n-1}\times R$ and show that in this case $M$ is necessarily a graph of a smooth function over $\Omega$. In particular, $M$ is non-singular.

Equations

$$w=Q(z)+O(|z{|}^3),$$

$$D=\Omega\times[m_{1},m_{2}]\Subset{\mathbb{B}}(r)\cap\big{(}\Omega\times i{\mathbb{R}}_{v}\big{)}$$

$$L_{(\zeta,\xi)}=H_{(\zeta,\xi)}+{\mathbb{R}}(0,1),\quad\Omega_{(\zeta,\xi)}=L_{(\zeta,\xi)}\cap\Omega,\quad S_{(\zeta,\xi)}=\big{(}H_{(\zeta,\xi)}+{\mathbb{C}}(0,1)\big{)}\cap S$$

$$\big{(}H_{(\zeta,\xi)}+{\mathbb{C}}(0,1)\big{)}\cap\big{(}{\rm b}\Omega\times i{\mathbb{R}}_{v}\big{)}$$

$$M_{(\zeta ,\xi )}=(H_{(\zeta ,\xi )}+\mathbb{C}(0,1))\cap M.$$

$${\mathcal{S}}_U=\bigcup _{q\in U}{\hat{S}}_q;$$

$$|(q_1,g(q_1))-(a_m,f(a_m))|\le |(q_1,g(q_1))-(b_m,g(b_m))|.$$

$$|(q_1,g(q_1))-(b_m,g(b_m))|\to 0$$

$${\Sigma }_m\cap {\pi }^{-1}(\overline{B})=\varnothing$$

$$V_{{\Sigma }_m}\cap {\pi }^{-1}(B)\ne \varnothing$$

Full text

In this paper we consider the case when $S$ is a graph of a smooth function over the boundary of a bounded strongly convex domain $\Omega\subset{\mathbb{C}}^{n-1}\times{\mathbb{R}}$ and show that in this case $M$ is necessarily a graph of a smooth function over $\Omega$. In particular, $M$ is non-singular.

*∗*Supported by the project MIUR ”Geometric Properties of Real and Complex Manifolds”.

*∗∗*Supported in part by the Science Foundation Ireland grant 06/RFP/MAT018.

1. Introduction

The problem of finding a Levi-flat hypersurface $M\subset{\mathbb{C}}^{n}$ with prescribed boundary $S$ (the complex analogue of the real Plateau’s problem), has been extensively studied for $n=2$ (cf. [Bi, BeG, BeK, Kr, CS, Sh, SlT, ShT]). In [DTZ2] (announced in [DTZ1]) we addressed this problem for $n\geq 3$, where the situation is substantially different. In contrast to the case $n=2$, for $n\geq 3$ the boundary $S$ has to satisfy certain compatibility conditions. Assuming those necessary conditions as well as the existence of complex points, their ellipticity and non-existence of complex subvarieties in $S$, we have constructed in [DTZ2] a (unique but possibly singular) solution to the above problem. An example was also provided in [DTZ2] showing that one may not always expect a smooth solution $M$ in general.

The purpose of this paper is to show that the solution $M$ is smooth if the given boundary has certain “graph form”. More precisely, in the coordinates $(z,u+iv)\in{\mathbb{C}}^{n-1}\times{\mathbb{C}}$, we assume that $S$ is the graph of a smooth function $g\colon{\rm b}\Omega\to{\mathbb{R}}_{v}$, where ${\rm b}\Omega$ is the smooth boundary of a strongly convex bounded domain $\Omega$ in ${\mathbb{C}}_{z}^{n-1}\times{\mathbb{R}}_{u}$ and $S$ satisfies the assumptions of [DTZ2] mentioned above. Let $M$ be the solution given by these theorems. Recall that it is obtained as a projection to ${\mathbb{C}}^{n}$ of a Levi-flat subvariety with negligible singularities in $[0,1]\times{\mathbb{C}}^{n}$. Let $q_{1},q_{2}\in{\rm b}\Omega$ be the projections of the complex points $p_{1},p_{2}$ of $S$. Using a theorem of Shcherbina on the polynomial envelope of a graph in ${\mathbb{C}}^{2}$ (cf. [Sh]) we here prove (cf. Theorem 3.1) that

• i)

the solution $M$ is the graph of a Lipschitz function $f\colon\overline{\Omega}\rightarrow{\mathbb{R}}_{v}$ with $f|_{{\rm b}\Omega}=g$ which is smooth on $\overline{\Omega}\setminus\{q_{1},q_{2}\}$;

• ii)

$M_{0}={\rm graph}(f)\smallsetminus S$ is a Levi flat hypersurface in ${\mathbb{C}}^{n}$.

The regularity of $f$ at $q_{1}$ and $q_{2}$ remains an interesting open problem closely related to the work of Kenig and Webster [KW1, KW2].

2. Preliminaries

In this section we collect some facts that will be used in the sequel.

2.1. Remarks about Harvey-Lawson theorem

Let $D$ be a strongly pseudoconvex bounded domain in ${\mathbb{C}}^{n}$, $n\geq 3$, with boundary ${\rm b}D$, $\Sigma\!\subset\!{\rm b}D$ a compact connected maximally complex $(2d-1)$-submanifold with $d>1$. Then, in view of the theorem of Harvey and Lawson in [HL1, Theorem 10.4] (see also [HL2]), $\Sigma$ is the boundary of a uniquely determined relatively compact subset $V\subset\overline{D}$ such that: $\overline{V}\setminus\Sigma$ is a complex analytic subset of $D$ with finitely many singularities of pure dimension $d$ and, near $\Sigma$, $\overline{V}$ is a $d$-dimensional complex manifold with boundary. We refer to $V=V_{\Sigma}$ as the solution of the boundary problem corresponding to $\Sigma$. A simple consequence is the following:

Lemma 2.1**.**

Let $D\subset{\mathbb{C}}^{n}$ be as above and $\Sigma_{1}$, $\Sigma_{2}$ connected, maximally complex $(2d-1)$-submanifolds of ${\rm b}D$. Let $V_{1}$, $V_{2}$ be the corresponding solutions of the boundary problem. If $d>1$, $2d>n$ and $\Sigma_{1}\cap\Sigma_{2}=\varnothing$, then $V_{1}\cap V_{2}=\varnothing$.

**Proof. **Suppose $V_{1}\cap V_{2}\neq\varnothing$. Then $2d>n$ implies $\dim V_{1}\cap V_{2}\geq 1$. Since $V_{1}\cap V_{2}$ is an analytic subset of $D$, its closure $\overline{V_{1}\cap V_{2}}$ must intersect ${\rm b}D$ and hence also $\Sigma_{1}\cap\Sigma_{2}\neq\varnothing$, which contradicts the assumption.  $\Box$

2.2. Known results

First, we have the following: a real $2$-codimensional submanifold $S$ of ${\mathbb{C}}^{n}$, $n\geq 3$, which locally bounds a Levi flat hypersurface must be nowhere minimal near a CR point, i.e. all local CR orbits must be of positive codimension (cf. [DTZ2, Section 2]). If $p\in S$ is a complex point, consider local holomorphic coordinates $(z,w)\in{\mathbb{C}}_{z}^{n-1}\times{\mathbb{C}}_{w}$, vanishing at $p$, such that $S$ is locally given by the equation

[TABLE]

where $Q(z)$ is a complex valued quadratic form in the real coordinates $({\sf Re}\,z,{\sf Im}\,z)\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}^{n-1}$. Observing that not all quadratic forms $Q$ can appear when $S$ bounds a Levi flat hypersurface one comes to the condition that $p$ must be flat, i.e. $Q(z)\in{\mathbb{R}}$ in suitable coordinates. A natural stronger condition is that of ellipticity which means by definition that $Q(z)\in{\mathbb{R}_{+}}$ for every $z\neq 0$ in suitable coordinates.

Assume that:

1. (1)

$S$ is compact, connected and nowhere minimal at its CR points; 2. (2)

$S$ has at least one complex point and every such point of is flat and elliptic; 3. (3)

$S$ does not contain complex manifold of dimension $(n-2)$.

Then in [DTZ2, Proposition 3.1] it was proved that

• a)

$S$ is diffeomorphic to the unit sphere with two complex points $p_{1},p_{2}$;

• b)

the CR orbits of $S$ are topological $(2n-3)$-spheres that can be represented as level sets of a smooth function $\nu:S\to{\mathbb{R}}$, inducing on $S_{0}=S\setminus\{p_{1},p_{2}\}$ a foliation $\mathcal{F}$ of class $C^{\infty}$ with $1$-codimensional compact leaves.

Next, by applying a parameter version of Harvey-Lawson’s theorem [HL1, Theorem 8.1], we obtained in [DTZ2, Theorem 1.3] a solution to the boundary problem as follows:

Theorem 2.2**.**

Let $S\subset{\mathbb{C}}^{n}$, $n\geq 3$ satisfy the above conditions. Then there exist a smooth submanifold $\widetilde{S}$ and a Levi flat $(2n-1)$-subvariety $\widetilde{M}$ in ${\mathbb{C}}^{n}\times[0,1]$ (i.e. $\widetilde{M}$ is Levi flat in ${\mathbb{C}}^{n}\times{\mathbb{C}}$) such that $\widetilde{S}=d\widetilde{M}$ in the sense of currents and the natural projection $\pi\colon{\mathbb{C}}^{n}\times[0,1]\to{\mathbb{C}}^{n}$ restricts to a diffeomorphism between $\widetilde{S}$ and $S$.

As for the singularities of $\widetilde{M}$ we have the following results [DTZ2, Theorems 1.4]:

Theorem 2.3**.**

The Levi-flat $(2n-1)$-subvariety $\widetilde{M}$ can be chosen with the following properties:

1. (i)

$\widetilde{S}$* has two complex points $\tilde{p}_{0}$ and $\tilde{p}_{1}$ with $\widetilde{S}\cap({\mathbb{C}}^{n}\times\{j\})=\{\tilde{p}_{j}\}$ for $j=0,1$; every other slice ${\mathbb{C}}^{n}\times\{x\}$ with $x\in(0,1)$, intersects $\widetilde{S}$ transversally along a submanifold diffeomorphic to a sphere that bounds (in the sense of currents) the (possibly singular) irreducible complex-analytic hypersurface $(\widetilde{M}\setminus\widetilde{S})\cap({\mathbb{C}}^{n}\times\{x\})$;* 2. (ii)

the singular set ${\rm Sing}\,\widetilde{M}$ is the union of $\widetilde{S}$ and a closed subset of $\widetilde{M}\setminus\widetilde{S}$ of Hausdorff dimension at most $2n-3$; moreover each slice $({\rm Sing}\,\widetilde{M}\setminus\widetilde{S})\cap({\mathbb{C}}^{n}\times\{x\})$ is of Hausdorff dimension at most $2n-4$; 3. (iii)

there exists a closed subset $\widetilde{A}\subset\widetilde{S}$ of Hausdorff $(2n-2)$-dimensional measure zero such that away from $\widetilde{A}$, $\widetilde{M}$ is a smooth submanifold with boundary $\widetilde{S}$ near $\widetilde{S}$; moreover $\widetilde{A}$ can be chosen such that each slice $\widetilde{A}\cap({\mathbb{C}}^{n}\times\{x\})$ is of Hausdorff $(2n-3)$-dimensional measure zero.

3. The case of graph

From now on we assume that $S\subset{\mathbb{C}}^{n}$, $n\geq 3$, is a graph. Consider ${\mathbb{C}}^{n}={\mathbb{C}}^{n-1}_{z}\times{\mathbb{C}}_{w}$ with complex coordinates $z=(z_{1},\ldots,z_{n-1})$ and $w$ where $z_{\alpha}=x_{\alpha}+iy_{\alpha}$, $1\leq\alpha\leq n-1$, $w=u+iv$. We also write {\mathbb{C}}^{n}=\big{(}{\mathbb{C}}^{n-1}_{z}\times{\mathbb{R}}_{u}\big{)}\times i{\mathbb{R}}_{v}. Accordingly, a point of ${\mathbb{C}}^{n}$ will be denoted by $(z,u,v)=(z,u+iv)$.

Let $\Omega$ be a bounded strongly convex domain of ${\mathbb{C}}^{n-1}_{z}\times{\mathbb{R}}_{u}$ with smooth boundary ${\rm b}\Omega$. By strong convexity here we mean that the second fundamental form of the boundary ${\rm b}\Omega$ of $\Omega$ is everywhere positive definite. In particular, $\Omega\times i{\mathbb{R}}_{v}$ is a strongly pseudoconvex domain in ${\mathbb{C}}^{n}$.

Let $g:{\rm b}\Omega\rightarrow{\mathbb{R}}_{v}$ be a smooth function, and $S\subset{\mathbb{C}}^{n}$ the graph of $g$. We assume that $S$ satisfies the conditions of [DTZ2, Theorem 1.3] and denote $q_{1},q_{2}\in{\rm b}\Omega$ the natural projections of the complex points $p_{1},p_{2}$ of $S$, respectively.

Our goal is to prove the following:

Theorem 3.1**.**

Let $q_{1},q_{2}\in{\rm b}\Omega$ be the projections of the complex points $p_{1},p_{2}$ of $S$, respectively. Then, there exists a Lipschitz function $f:\overline{\Omega}\rightarrow{\mathbb{R}}_{v}$ which is smooth on $\overline{\Omega}\setminus\{q_{1},q_{2}\}$ and such that $f_{|{\rm b}\Omega}=g$ and $M_{0}={\rm graph}(f)\smallsetminus S$ is a Levi flat hypersurface of ${\mathbb{C}}^{n}$. Moreover, each complex leaf of $M_{0}$ is the graph of a holomorphic function $\phi\colon\Omega^{\prime}\to{\mathbb{C}}$ where $\Omega^{\prime}\subset{\mathbb{C}}^{n-1}$ is a domain with smooth boundary (that depends on the leaf) and $\phi$ is smooth on $\overline{\Omega}^{\prime}$.

The natural candidate to be the graph $M$ of $f$ is \pi\big{(}\widetilde{M}\big{)} where $\widetilde{M}$ and $\pi$ are as in Theorem 2.2. We prove that this is the case proceeding in several steps.

3.1. Behaviour near $S$

Set $m_{1}=\min\limits_{S}g$, $m_{2}=\max\limits_{S}g$ and $r\gg 0$ such that

[TABLE]

where ${\mathbb{B}}(r)$ is the ball $\{|(z,w)|<r\}$.

Let $\Sigma$ be a CR-orbit of the foliation of $S\setminus\{p_{1},p_{2}\}$. Then, $\Sigma$ is a compact maximally complex $(2n-3)$-dimensional real submanifold of ${\mathbb{C}}^{n}$, which is contained in the boundary of the strongly pseudoconvex domain $\Omega\times i{\mathbb{R}}_{v}$ of ${\mathbb{C}}^{n}$. Let $V$ be the solution to the boundary problem corresponding to $\Sigma$, i.e. the complex-analytic subvariety of $\Omega\times i{\mathbb{R}}_{v}$ bounded by $\Sigma$. We refer to $V$ as the leaf bounded by $\Sigma$. From Theorems 2.2 and 2.3 it follows that $V$ is obtained as projection $\pi(\widetilde{V})$, where $\widetilde{V}=(\widetilde{M}\setminus\widetilde{S})\cap({\mathbb{C}}^{n}\times\{x\})$ for suitable $x\in(0,1)$. In particular, if $M$ denotes $\pi(\widetilde{M})$, $\pi_{|\widetilde{V}}$ defines a biholomorphism $\widetilde{V}\simeq V$ and $M\setminus S\subset D$.

Now let $\Sigma_{1}$ and $\Sigma_{2}$ be two distinct CR orbits of the foliation of $S\setminus\{p_{1},p_{2}\}$, and let $\overline{V}_{1}$, $\overline{V}_{2}$ be the corresponding leaves bounded by them. Then $\overline{V}_{1}$, $\overline{V}_{2}$ do not intersect by Lemma 2.1.

Remark 3.1**.**

In the previous discussion, we only employed the fact that $\Omega\times{\mathbb{R}}_{v}$ is a strongly pseudoconvex domain and $S$ is contained in its boundary, without regarding the graph nature of $S$. It can happen that the leaves have isolated singularities. We shall show that this cannot happen in our case.

Lemma 3.2**.**

Let $p\in S$ be a CR point. Then, near $p$, $M$ is the graph of a function $\phi$ on a domain $U\!\subset\!{\mathbb{C}}_{z}^{n-1}\times{\mathbb{R}}_{u}$, which is smooth up to the boundary of $U$.

**Proof. **Near $p$, $S$ is foliated by local CR orbits. As a consequence of Theorem 2.2, each local CR orbit extends to a compact global CR orbit $\Sigma$ that bounds a complex codimension $1$ subvariety $V_{\Sigma}\subset\Omega\times i{\mathbb{R}}_{v}$. Furthermore, near $p$, each $\Sigma$ is smooth and can be represented as the graph of a CR function over a strongly pseudoconvex hypersurface and $V_{\Sigma}$ as the graph of the local holomorphic extension of this function. It follows from the Hopf Lemma that $V$ is transversal to the strongly pseudoconvex hypersurface $b\Omega\times i{\mathbb{R}}_{v}$ near $p$. Hence the family of $V_{\Sigma}$ near $p$ forms a smooth real hypersurface with boundary on $S$ that can be seen as the graph of a smooth function $\phi$ from a relative open neighbourhood $U$ of $p$ in $\overline{\Omega}$ into ${\mathbb{R}}_{v}$. Finally, Lemma 2.1 guarantees that this family does not intersect any other leaf $V$ from $M$. This completes the proof.  $\Box$

Corollary 3.3**.**

If $p\in S$ is a CR point, each complex leaf $V$ of $M$, near $p$, is the graph of a holomorphic function on a domain $\Omega_{V}\subset{\mathbb{C}}^{n-1}_{z}$, which is smooth up to the boundary of $\Omega_{V}$.

**Proof. **Since $M$ is the graph of a smooth function near $p$, its tangent space at every point near $p$ is transversal to $i{\mathbb{R}}_{v}$. Hence the complex tangent space of $M$ at every point near $p$ is transversal to ${\mathbb{C}}_{w}$. Since the tangent spaces of the complex leaves of $M$ coincide with the complex tangent spaces of $M$, it follows that each leaf $V$ projects immersively to ${\mathbb{C}}^{n-1}_{z}$ and the conclusion follows.  $\Box$

3.2. $M$ is the graph of a Lipschitz function.

Assume as before that $\Omega$ is strongly convex. We have the following

Proposition 3.4**.**

$M$* is the graph of a Lipschitz function $f:\overline{\Omega}\to{\mathbb{R}}_{v}$.*

**Proof. **We fix a nonzero vector $a\in{\mathbb{C}}^{n-1}_{z}$ and for a given point $(\zeta,\xi)\in\Omega$ denote by $H_{(\zeta,\xi)}\subset{\mathbb{C}}_{z}^{n-1}\times\{\xi\}$ the complex line through $(\zeta,\xi)$ in the direction of $(a,0)$. Furthermore, we set

[TABLE]

Then $S_{(\zeta,\xi)}$ is contained in the strongly convex cylinder

[TABLE]

over $H_{(\zeta,\xi)}+{\mathbb{C}}(0,1)\simeq{\mathbb{C}}^{2}$ and it is the graph of $g_{|{\rm b}\Omega_{(\zeta,\xi)}}$.

Since $\Omega_{(\zeta,\xi)}=\Omega\cap L_{(\zeta,\xi)}$, in view of the main theorem of [Sh], the polynomial hull $\widehat{S}_{(\zeta,\xi)}$ of $S_{(\zeta,\xi)}$ is a continuous graph over ${\overline{\Omega}}_{(\zeta,\xi)}$. Consider $M=\pi(\widetilde{M})$ and set

[TABLE]

Since $M$ is a union of irreducible analytic subvarieties of codimension $1$ in ${\mathbb{C}}^{n}$ with boundary in the graph $S$, each intersection $M_{(\zeta,\xi)}$ is the union of a family $\mathcal{A}$ of $1$-dimensional analytic subsets. Clearly, the boundary of a connected component of any such analytic set is contained in $S_{(\zeta,\xi)}$. It follows that $M_{(\zeta,\xi)}$ is contained in the polynomial hull $\widehat{S}_{(\zeta^{\prime},\xi)}$ of $S_{(\zeta,\xi)}$. In view of the main theorem of Shcherbina [Sh], $\widehat{S}_{(\zeta,\xi)}$ is a graph over ${\overline{\Omega}}_{(\zeta,\xi)}=\overline{\Omega}\cap L_{(\zeta,\xi)}$, foliated by analytic discs, so $M_{(\zeta,\xi)}$ is a graph over a subset $U$ of ${\overline{\Omega}}_{(\zeta,\xi)}$.

On the other hand, every analytic disc $\Delta$ of $\widehat{S}_{(\zeta,\xi)}$ has its boundary on $S_{(\zeta,\xi)}\subset S$. Since all elliptic complex points are isolated, the boundary of $\Delta$ contains a CR point $p$ of $S$. In view of Lemma 3.2, near $p$, $M_{(\zeta,\xi)}$ is also a graph over ${\overline{\Omega}}_{(\zeta,\xi)}$. Thus, near $p$, we must have $M_{(\zeta,\xi)}=\widehat{S}_{(\zeta,\xi)}$. In particular, near $p$, $\Delta$ is contained in $M_{(\zeta,\xi)}$, and therefore in a leaf $V_{\Sigma}$ of $M$. Since $V_{\Sigma}$ is a closed analytic subset in ${\mathbb{C}}^{n}\setminus S$, the whole disc $\Delta$ is contained in $V_{\Sigma}$ and hence in $M$. Moreover, $\Delta\subset H_{(\zeta,\xi)}+{\mathbb{C}}(0,1)$ thus we conclude that $\Delta\subset M_{(\zeta,\xi)}$. Therefore, every analytic disc of $\widehat{S}_{(\zeta,\xi)}$ is contained in $M_{(\zeta,\xi)}$, consequently $M_{(\zeta,\xi)}$ and $\widehat{S}_{(\zeta,\xi)}$ coincide. It follows that $M$ is the graph of a function $f:\overline{\Omega}\to{\mathbb{R}}_{u}$.

Let us prove that $f$ is a continuous function. Choose $(\zeta,\xi)\in\Omega$ and a complex line $H_{(\zeta,\xi)}$ as before. Consider a neighborhood $U$ of $(\zeta,\xi)$ in ${\mathbb{C}}_{z}^{n-1}\times{\mathbb{R}}_{u}$. For $q\in U$, let $H_{q}$ be the translated of $H_{(\zeta,\xi)}$ which passes through $q$. With the notation corresponding to the one employed above, we can state the following. For a small enough neighborhood $V\subset U$ of $p$ in ${\mathbb{C}}_{z}^{n-1}\times{\mathbb{R}}_{u}$, let $\widehat{S}_{q}$ be the polynomial hull of $S_{q}$ in $H_{q}+{\mathbb{C}}(0,1)$, and let

[TABLE]

then $\mathcal{S}_{U}$ is the graph of a continuous function. Indeed let $\overline{q}$ be a point in $V$, and let $\{q_{m}\}_{m\in{\mathbb{N}}}$ be a sequence of points such that $q_{n}\to\overline{q}$. Then, obviously, the sets $S_{q_{m}}$ converge to the set $S_{\overline{q}}$ in the Hausdorff metric as $n\to\infty$. Moreover, it is also clear that $\widetilde{\Omega}_{q_{n}}\to\widetilde{\Omega}_{\overline{q}}$ for $n\to\infty$. Then, by [Sh, Lemma 2.4] it follows that $\widehat{S}_{q_{m}}\to\widehat{S}_{\overline{q}}$ as $m\to\infty$. Since every $\widehat{S}_{q}$ is a continuous graph, this allows to prove easily that $\mathcal{S}_{U}$ is a continuous graph as a whole.

Thus, $f$ is continuous on $\Omega$, whence on $\overline{\Omega}\smallsetminus\{q_{1},q_{2}\}$ in view of Lemma 3.2. Continuity at $q_{1}$ is proved as follows. Let let $\{a_{m}\}_{m\in{\mathbb{N}}}\!\subset\!\Omega$ be a sequence of points which converges to $q_{1}$. Each point \big{(}a_{m},f(a_{m})\big{)} belongs to a complex leaf $V_{\Sigma_{m}}$ of $M$ which is bounded by a compact CR orbit $\Sigma_{m}$ of the foliation of $S\smallsetminus\{p_{1},p_{2}\}$ (cf. Section 2). By the maximum principle, for every $m\in{\mathbb{N}}$ there exists a point $(b_{m},g(b_{m}))$ in $\Sigma_{m}$ such that

[TABLE]

We claim that

[TABLE]

as $m\to\infty$. If not there exists an open $B=B(q_{1},r)\Omega\times{\mathbb{R}}_{u}$ centered at $q_{1}$ such that $b_{m}\notin\overline{B}$ for all $m$. It follows that

[TABLE]

for all $m$ and

[TABLE]

for $m\gg 0$. This violates the Kontinuitätsatz since $\Omega\times i{\mathbb{R}}_{v}$ is a domain of holomorphy.

Continuity at $q_{2}$ is proved in a similar way.

Thus $f$ is continuous on $\overline{\Omega}$ and smooth near ${\rm b}\Omega\smallsetminus\{q_{1},q_{2}\}$.

In order to show that $f$ is Lipschitz we now observe that, as it is easily proved, $f_{|\Omega}$ is a weak solution of the Levi-Monge-Ampère operator defined in [SlT] with smooth boundary value, so, in view of [SlT, Theorems 2.4, 4.4, 4.6], it is Lipschitz. This concludes the proof of Proposition 3.4.  $\Box$

Remark 3.2**.**

$M$ is the envelope of holomorphy of $S$.

3.3. Regularity.

In order to prove that $M\smallsetminus\{p_{1},p_{2}\}$ is a smooth manifold with boundary we need the following:

Lemma 3.5**.**

Let $U$ be a domain in ${\mathbb{C}}^{n-1}_{z}\times{\mathbb{R}}_{u}$, $n\geq 2$, $f:U\to{\mathbb{R}}_{v}$ a continuous function. Let $A\!\subset\!{\rm graph}(f)$ be a germ of complex analytic set of codimension $1$. Then $A$ is a germ of a complex manifold, which is a graph over ${\mathbb{C}}^{n-1}_{z}$.

**Proof. **The idea of the proof (here is slightly modified) is due to Jean-Marie Lion cfr. [L].

Let us denote by $z_{1},\ldots,z_{n-1},w=u+iv,$ the complex coordinates in ${\mathbb{C}}^{n-1}_{z}\times{\mathbb{C}}_{w}$. We may suppose that $A$ is a germ at [math]. Let $h\in\mathcal{O}_{n+1}$ be a non identically vanishing germ of holomorphic function such that $A=\{h=0\}$. Let ${\mathbb{D}}_{\varepsilon}$ be the disc $\{z=0\}\cap\{|w|<\varepsilon\}$. Then, for $\varepsilon<<1$, we have either $A\cap{\mathbb{D}}_{\varepsilon}=\{0\}$ or $A\cap{\mathbb{D}}_{\varepsilon}={\mathbb{D}}_{\varepsilon}$. The latter is not possible since ${\mathbb{D}}_{\varepsilon}$ is not contained in any graph over ${\mathbb{C}}^{n-1}\times{\mathbb{R}}_{u}$. It follows that $A\cap{\mathbb{D}}_{\varepsilon}=\{0\}$, i.e. $A$ is $w$-regular. Let us denote by $\pi$ the projection ${\mathbb{C}}^{n}_{z}\to{\mathbb{C}}^{n-1}_{z}$. Then, by the local parametrization theorem for analytic sets there exists $d\in{\mathbb{N}}$ such that

• •

for some neighborhood $U$ of [math] in ${\mathbb{C}}^{n-1}_{z}$, there exists an analytic set $\Delta\subset U$ such that $A_{\Delta}=A\cap((U\setminus\Delta)\times{\mathbb{D}}_{\varepsilon})$ is a manifold;

• •

$\pi:A_{\Delta}\to U\setminus\Delta$ is a $d$-sheeted covering.

We claim that the covering $\pi:H_{\Delta}\to U\setminus\Delta$ is trivial. Otherwise, there would exist a closed loop $\gamma:[0,1]\to U\setminus\Delta$ whose lift $\widetilde{\gamma}$ to $A_{\Delta}$ is not closed. We extend $\gamma$ to ${\mathbb{R}}$ by periodicity and extend $\widetilde{\gamma}$ to ${\mathbb{R}}$ as lift of $\gamma$. Define $\alpha=u\circ\widetilde{\gamma}=u\circ\gamma$, $\beta=v\circ\widetilde{\gamma}$. Since $\alpha$ is continuous and bounded, there exists $\theta\in{\mathbb{R}}$ such that $\alpha(\theta)=\alpha(\theta+1)$. But then $\beta(\theta)=\beta(\theta+1)$ since by the assumption, $\beta(\theta)=f(\gamma(\theta),\alpha(\theta))$. Hence $\widetilde{\gamma}(\theta)=\widetilde{\gamma}(\theta+1)$, a contradiction with the assumption that $\widetilde{\gamma}$ is not closed.

Since $\pi:A_{\Delta}\to U\setminus\Delta$ is a trivial covering, we may define $d$ holomorphic functions $\tau_{1},\ldots,\tau_{d}:U\setminus\Delta\to{\mathbb{C}}$ such that $A_{\Delta}$ is a union of the graphs of the $\tau_{j}$’s. By Riemann’s extension theorem, the functions $\tau_{j}$ extend as holomorphic functions $\tau_{j}\in\mathcal{O}(U)$. The desired conclusion will follow from the fact that all the $\tau_{j}$ coincide. Indeed, suppose, by contradiction, $\tau_{1}\neq\tau_{2}$; then for some disc ${\mathbb{D}}\subset U$ centered at [math] we have $\tau_{1}|_{\mathbb{D}}\neq\tau_{2}|_{\mathbb{D}}$ and then, after shrinking ${\mathbb{D}}$, $(\tau_{1}-\tau_{2})|_{\mathbb{D}}$ vanishes only at [math]. But, by virtue of the hypothesis, $\{{\sf Re}\,(\tau_{1}-\tau_{2})=0\}\subset\{\tau_{1}-\tau_{2}=0\}=\{0\}$, when restricted to ${\mathbb{D}}$. The latter is not possible since $(\tau_{1}-\tau_{2})|_{{\mathbb{D}}}\neq 0$ is holomorphic and thus an open map (whose image must include a segment of the imaginary axis).  $\Box$

Proof of Theorem 3.1. Consider the foliation on $S\smallsetminus\{p_{1},p_{2}\}$ given by the level sets of the smooth function $\nu\colon S\to[0,1]$ as in Section 2 and set $L_{t}=\{\nu=t\}$ for $t\in(0,1)$. Let $V_{t}\subset\overline{\Omega}\times i{\mathbb{R}}_{v}\subset{\mathbb{C}}^{n}$ be the complex leaf of $M$ bounded by $L_{t}$ and $\pi\colon{\mathbb{C}}^{n-1}_{z}\times{\mathbb{C}}_{w}\to{\mathbb{C}}_{z}^{n-1}$ denote the natural projection. We have:

• •

by Proposition 3.4, $M$ is the graph of a continuous function over $\Omega$ and by Lemma 3.5, each leaf $V_{t}$ is a complex hypersurface and $\pi|_{V_{t}}$ is a submersion.

• •

Since $\Omega$ is strongly convex, an argument completely analogous to that of [Sh, Lemma 3.2] shows that $\pi_{|V_{t}}$ is one-to-one, then, by Corollary 3.3, $\pi$ sends $V_{t}$ onto a domain $\Omega_{t}\subset{\mathbb{C}}^{n-1}_{z}$ with smooth boundary.

If

• \pi_{u}:\big{(}{\mathbb{C}}^{n-1}_{z}\times{\mathbb{R}}_{u}\big{)}\times i{\mathbb{R}}_{v}\to{\mathbb{R}}_{u},
• \pi_{v}:\big{(}{\mathbb{C}}^{n-1}_{z}\times{\mathbb{R}}_{u}\big{)}\times i{\mathbb{R}}_{v}\to{\mathbb{R}}_{v}

denote the natural projections then ${\pi_{u}}_{|L_{t}}=a_{t}\circ\pi_{|L_{t}}$ and ${\pi_{v}}_{|L_{t}}=b_{t}\circ\pi_{|L_{t}}$, where $a_{t}$ and $b_{t}$ are smooth functions in ${\rm b}\Omega_{t}$. Furthermore, the boundary ${\rm b}\Omega_{t}$ and $a_{t}$, $b_{t}$ depend smoothly on $t$ for $t\in(0,1)$. The latter property means that one has a local parametrization of ${\rm b}\Omega_{t}$ smoothly depending on $t$ and such that the functions $a_{t}$, $b_{t}$ also depend smoothly on $t$ when composed with this parametrization. It follows that

• •

if $(z_{t},w_{t})\in M$, then $w_{t}=u_{t}+iv_{t}$ is varying in $V_{t}$, so $u_{t}+iv_{t}$ is the holomorphic extension to $\Omega_{t}$ of $a_{t}+ib_{t}$. In particular, $u_{t}$ and $v_{t}$ are smooth functions in $(z,t)$, e.g. as a consequence of the Martinelli-Bochner formula.

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The derivative $\partial u_{t}/\partial t$ is defined and harmonic in $\Omega_{t}$ for each $t$, and has a smooth extension to the boundary ${\rm b}\Omega_{t}$. Moreover, it follows from Lemma 3.2 and Corollary 3.3 that $\partial u_{t}/\partial t$ does not vanish on ${\rm b}\Omega_{t}$. Since the CR orbits $L_{t}$ are connected in view of Theorem 2.2, the boundary ${\rm b}\Omega_{t}$ is also connected and hence $\partial u_{t}/\partial t$ has constant sign on ${\rm b}\Omega_{t}$. Then, by the maximum principle, $\partial u_{t}/\partial t$ has constant sign in $\Omega_{t}$ and, in particular, does not vanish. The latter implies the $M\setminus S$ is the graph of a smooth function over $\Omega$, which extends smoothly to $\overline{\Omega}\setminus\{q_{1},q_{2}\}$.

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It furthermore follows from Proposition 3.4 that $M$ is the graph of a Lipschitz function over $\overline{\Omega}$. This completes the proof of Theorem 3.1.