SEMI LOCAL LEVI-FLAT EXTENSIONS

NIKOLAY SHCHERBINA^{∗} AND GIUSEPPE TOMASSINI^{∗∗}

Contents

1. Introduction. 1 2. Preliminaries. 5 3. Some properties of E(U). 6 4. Characterization of the CR-hull of rigid surfaces. 10 5. Proof of Theorems 3 and 4. 16 6. On maximality of E(U). 22 7. Examples. 24 References 29

1. Introduction.

Let G be a domain in ℂ_{z} × ℝ_{u} ⊂ℂ_{z,w}^{2} (w = u + iv). Let φ : b G → ℝ_{v} be a continuous function and Γ(φ) its graph. The extendability of φ to a continuous function Φ : \overline{G} → ℝ_{v} with the Levi-flat (i.e. foliated by holomorphic curves) graph Γ(Φ) has been studied by several authors ([BG], [BK], [E], [Kr], [Sh], [CS] in the case G is bounded, and [ST] in the unbounded case) under the assumption that bG is smooth and strictly pseudoconvex (i.e. G × ℝ_{v} is a strictly pseudoconvex domain of ℂ_{z,w}^{2}). In this paper we study a semi-local version of the extension problem, namely, when the function φ is prescribed on an open subset U of bG, where bG is smooth and strictly pseudoconvex. Namely, we define a notion of the hull E(U) ⊂ \overline{G} of U and prove that every continuous function φ : U → ℝ_{v} has a Levi-flat extension to E(U).

Date: December 4, 2003.

1991 Mathematics Subject Classification. Primary 35H05, 32Q60, 32W50 57R30, 53C38; Secondary 35H05.

∗Partially supported by Jubileumsfonden of the University of Göteborg.

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

Given an open subset 𝔻 of ℂ^{n} we denote A(𝔻) the Fréchet algebra O(D) ∩ C^{0}(\overline{𝔻}). If K is a compact subset of \overline{𝔻} let us denote by K̂_{A(𝔻)} (or simply by K̂) the A(𝔻)-hull of K.

If 𝔻 is a strictly pseudoconvex domain in ℂ^{n} and V is an open subset of b𝔻, we define the set E(V) to be equal to the union of the hulls K̂_{A(𝔻)} for all compact subsets K of V. It is easy to see that if K_{1} ⊂ K_{2} ⊂ K_{3} ⊂ ⋯ is an exhaustion of V by compact subsets, then E(V) = ⋃_{n} K̂_{n}. Denote also by E(V) the CR-hull of V. Since the

domain 𝔻 is assumed to be strictly pseudoconvex, CR-functions defined on V admit holomorphic extension to a one-sided neighbourhood V of V in the closed domain \overline{𝔻} and then E(V) is just the union of V with the hull of holomorphy of the interior part of V.

The first result of this paper describes some properties of the set E(V).

Theorem 1. Let 𝔻 be a strictly pseudoconvex domain in ℂ_{z,w}^{n+1}, where z = (z_{1}, z_{2}, ..., z_{n}), w = u + iv, and let V be an open subset of b𝔻. Then

1) E(V) is an open subset of \overline{𝔻} such that E(V) ∩ b𝔻 = V and E(V) ∩ 𝔻 is pseudoconvex.

If 𝔻 is of the form 𝔻 = G × ℝ_{v}, where G is a domain in ℂ_{z}^{n} × ℝ_{u}, and V is of the form V = U × ℝ_{v}, where U ⊂bG, then

2) E(V) is invariant under translation in v-direction. In particular, there is an open subset E(U) of \overline{G} such that E(V) = E(U) × ℝ_{v}.

If, moreover, n = 1 and U is the union of simply-connected subdomains of bG, then

3) E(U × ℝ_{v}) = E(U × ℝ_{v}).

In the case when G is a topological 3-ball the set E(U) enjoies the following property.

Theorem 2. Let G ⊂ ℂ_{z} × ℝ_{u} be a domain diffeomorphic to a 3-ball such that G × ℝ_{v} ⊂ ℂ_{z,w}^{2} is strictly pseudoconvex. Let U be an open subset of bG and let {U_{α}} be the connected components of U. Then the sets E(U_{α}) are connected and disjoint and, moreover, E(U) = ⋃_{α} E(U_{α}).

Note that the statement of Theorem 2 is in general false if G is not diffeomorphic to a 3-ball, as it is shown by Example 1 in Section 7.

In view of part 3 in Theorem 1 one can raise the following question: How to describe the CR-hull of the surface U × ℝ_{v} ? A general construction of the hull of holomorphy of a rigid domain was given in the classical paper [Br]. This result was generalised in [CS], [Ki] and [TT]. Since in our case the surface U × ℝ_{v} is already contained in the boundary of strictly pseudoconvex domain G × ℝ_{v}, it is possible to provide a concrete geometric description of its CR-hull. Such description is given in Section 4 of this paper and is used later in Example 2 presented in Section 7.

Let, as above, G ⊂ ℂ_{z} × ℝ_{u} be a domain such that G × ℝ_{v} ⊂ ℂ_{z,w}^{2} is strictly pseudoconvex, U an open subset of bG and φ : U → ℝ_{v} a continuous function. Consider the sets

V_{φ}^{+} = {(z, w)∈U × ℝ_{v} : φ(z, u) < v}

and

V_{φ}^{−} = {(z, w)∈U × ℝ_{v} : v < φ(z, u)}.

Then the following theorem holds.

Theorem 3. The sets E(V_{φ}^{±}) can be represented in the following form:

(1) E(V_{φ}^{+}) = \{(z, w)∈\overline{G} × ℝ_{v} : v > Φ^{+}(z, u), (z, u)∈E(U)\},
(2) E(V_{φ}^{−}) = \{(z, w)∈\overline{G} × ℝ_{v} : v < Φ^{−}(z, u), (z, u)∈E(U)\},

for some functions Φ^{±} defined on the set E(U).

Having defined the functions Φ^{±}, we can formulate the main result of this paper.

Theorem 4. Let G be a domain in ℂ_{z} × ℝ_{u} such that G × ℝ_{v} ⊂ ℂ_{z,w}^{2} is strictly pseudoconvex. Let U be an open subset of bG and E(U) the defined above open subset of \overline{G}. Then for every φ ∈ C(U) there exist two functions Φ^{±} ∈ C(E(U)) such that Φ^{±}|_{U} = φ and their graphs Γ(Φ^{±}) are Levi-flat over E(U) ∩ G.

Moreover, for any function Φ∈C(E(U)) such that Φ|_{U} = φ and Γ(Φ) is Levi-flat over E(U) ∩ G one has

Φ^{−}(z, u) ≤ Φ(z, u) ≤ Φ^{+}(z, u)

for each point (z, u)∈E(U).

For bounded domains G ⊂ ℂ_{z} × ℝ_{u} such that G × ℝ_{v} is strictly pseudoconvex the following characterization was given in Theorem 1 of [CS]: there is a domain G over ℂ_{z} and functions h^{−} and −h^{+} continuous in \overline{G}, smooth and strictly subharmonic in G, such that

G = {(z, u) : z = π(ζ), h^{−}(ζ) < u < h^{+}(ζ), ζ ∈ G},

where π : G → ℂ_{z} is the projection. It is easy to see that the same characterization holds true for unbounded domains G with the only difference that one of the functions h^{−} of −h^{+} can be identically equal to −∞. Following the notations in [CS] we denote by

bG^{±} = {(z, u) : z = π(ζ), u = h^{±}(ζ), ζ ∈ G}

respectively the “upper” and the “lower” part of bG (if h^{−} ≡ −∞ or h^{+} ≡ +∞, then the respective set bG^{−} or bG^{+} is empty).

Now we can formulate our result on maximality of E(U).

Theorem 5. Let G⊂ℂ_{z} × ℝ_{u} be a domain such that G × ℝ_{v} is strictly pseudoconvex. Let U be an open subset of bG such that each connected component U_{α} of U can be represented as U_{α} = U_{α}′ ∖ K_{α}, where U_{α}′ is a simply-connected subdomain of either bG^{−} or bG^{+} and K_{α} is a compact subset of U_{α}′. Then there is a function φ ∈ C(U) which does not admit a continuous Levi-flat extension to any domain Ω ⊂ \overline{G} such that E(U) ⊊ Ω.

Remark. It is easy to see that in the case when the domain G is diffeomorphic to a 3-ball it is enough to assume that each connected component U_{α} of U is contained either in bG^{−} or bG^{+}.

Remark. The main part of our results holds true for domains G with much weaker regularity assumptions as in [CS] and [ST]. We present them for strictly pseudoconvex case to avoid additional technical difficulties.

Acknowledgements. Part of this work was done while the first author was a visitor at Scuola Normale Superiore (Pisa) and at the Max Planck Institute of Mathematics (Bonn). It is his pleasure to thank both institutions for their hospitality and excellent working conditions.

2. Preliminaries.

Lemma 1. Let 𝔻 be an unbounded strictly pseudoconvex domain in ℂ^{n}. Let K be a compact subset of b𝔻. Then there exists a bounded strictly pseudoconvex subdomain 𝔻′ of 𝔻 such that K ⊂ b𝔻′ and the A(𝔻)-hull K̂_{A(𝔻)} is equal to the A(𝔻′)-hull K̂_{A(𝔻′)} of K. In particular, in the case n = 2, this implies that the set 𝔻 ∖ K̂_{A(𝔻)} is pseudoconvex.

Proof. Let 𝔹_{R} = {z ∈ ℂ^{n} : |z| < R}. Since K̂_{A(𝔻)} is a compact subset of \overline{𝔻}, we can choose R ∈ ℝ sufficiently large so that K̂_{A(𝔻)}⊂𝔹_{R} ∩\overline{𝔻} and b𝔹_{R} intersects b𝔻 transversally. Let us set 𝔻_{1} = 𝔹_{R} ∩ 𝔻. Take r < R such that the inclusion K̂_{A(𝔻)} ⊂ 𝔹_{r} ∩ \overline{𝔻} still holds true and consider a neighbourhood V of b𝔹_{R} ∩ b𝔻 such that V ∩ \overline{𝔹}_{r} = ∅. b𝔻_{1} is piece-wise smooth so, given an open neighbourhood V of b𝔹_{R} ∩ b𝔻, there exists a strictly pseudoconvex domain 𝔻′ with the properties 𝔻′ ⊂ 𝔻_{1} and 𝔻_{1} ∖ 𝔻′ ⊂ V (see e.g. [T]). We claim that 𝔻′ satisfies the properties of Lemma 1. Indeed, since A(𝔻) is a subalgebra of A(𝔻′), one has K̂_{A(𝔻′)}⊂ K̂_{A(𝔻)}. Assume, by a contradiction, that K̂_{A(𝔻)} ∖ K̂_{A(𝔻′)} ≠ ∅ and let P ∈ K̂_{A(𝔻)} ∖ K̂_{A(𝔻′)}. Then, by the definition of hull, there is a function f ∈ A(𝔻′) such that |f(P)| > 1 and ||f||_{K} < 1. We are going to approximate f by functions of A(𝔻) what will provide the desired contradiction. In view of a theorem of Kerzman [K], for every ε > 0 there exists a function g holomorphic in a neighbourhood W of \overline{𝔻}′ such that ||g − f||\overline{𝔻}′ < ε. Now we use the fact that \overline{𝔻} has a basis of Stein neighbourhoods [T] to choose a Stein domain U ⊃ \overline{𝔻} such that U ∩ 𝔹_{r} ⋐W. Since U ∩ 𝔹_{r} is a sublevel set in U of a plurisubharmonic function ρ(z) = |z|^{2}, we know by a theorem of Narasimhan [N] that U ∩𝔹_{r} is Runge in U. Hence, by the choice of r, there exists a function g̃ holomorphic in U such that ||g̃ − g||_{K∪{P }} < ε. It follows now that g_{1} = g̃|\overline{𝔻} satisfies ||g_{1} − f||_{K∪\{P \}} < 2ε. Thus, if ε is chosen sufficiently small, g_{1} satisfies the same condition as f, |g_{1}(P)| > 1 and ||g_{1}||_{K} < 1 which gives us a contradiction.

The last statement of lemma on pseudoconvexity of 𝔻∖K̂_{A(𝔻)} follows directly from strict pseudoconvexity of 𝔻 and from the fact, which can be found for example in [St], that in the case n = 2 the set K̂_{A(𝔻′)} is pseudoconcave in 𝔻′. □

3. Some properties of E(U).

Proof of Theorem 1. 1) Let P be a point of the set E(V). Consider an exhaustion K_{1} ⊂ K_{2} ⊂ K_{3} ⊂ ⋯ of V by compact subsets such that for each n = 1, 2, ... the set K_{n} is a subset of the interior part of K_{n+1}. Then, by definition of E(V), P ∈K̂_{n} for some n∈ℕ. Since 𝔻 is strictly pseudoconvex, it follows that there is a one-sided neighbourhood V_{n+1} of the interior part of K_{n+1} (relatively to b𝔻) in \overline{𝔻} filled by holomorphic discs with boundary on K_{n+1}. Consider δ_{0} > 0 such that for any vector e of length 1 one has (K_{n} + δe) ∩ \overline{𝔻}⊂V_{n+1} and (K̂_{n} + δe) ∩ b𝔻⊂K_{n+1} for all δ, 0 ≤ δ < δ_{0}. For any such δ consider the set E_{n,δ} = (K̂_{n} +δe)∩ \overline{𝔻}. By Lemma 1, the set 𝔻 ∖ K̂_{n} is pseudoconvex, consequently the set E_{n,δ} is pseudoconcave in 𝔻 ∖ V_{n+1} and, therefore, it is contained in the A(𝔻)-hull ˆ \overline{V}_{n+1} of \overline{V}_{n+1}. (This is clear because if not, there exist f ∈ A(𝔻) and Q ∈ E_{n,δ} such that |f| < 1 on E_{n,δ} ∩ \overline{V}_{n+1} and |f(Q)| = 1. Consider the function ϱ = log |f| + ε(|z|^{2} + |w|^{2}); ϱ is strictly plurisubharmonic in 𝔻\{f = 0} and, if ε is small enough, then for some sublevel set L_{a} = \{ϱ < a\} we have E_{n,δ} ∩ \overline{V}_{n+1} ⋐L_{a}, E_{n,δ} ⊂\overline{L}_{a} and E_{n,δ} ∩ bL_{a} ≠ ∅ which contradicts the pseudoconcavity of E_{n,δ}). Therefore,

E_{n,δ} ⊂ ˆ\overline{V}_{n+1} ⊂K̂_{n+1}.

(The inclusion ˆ \overline{V}_{n+1} ⊂K̂_{n+1} is a consequence of the maximum principle owing the fact that V_{n+1} is filled by holomorphic discs with boundary on K_{n+1}.) Hence, if P + δe ∈ \overline{𝔻}, one has P + δe ∈ K̂_{n+1} ⊂ E(V) for all δ such that 0 ≤ δ < δ_{0}. This proves that E(V) is an open subset of \overline{𝔻}.

The property E(V) ∩ b𝔻 = V follows directly from strict pseudoconvexity of 𝔻 and the definition of E(V). Pseudoconvexity of the set E(V) ∩ 𝔻 is the consequence of an obvious fact that the interior intK̂_{n}

of each set K̂_{n} is pseudoconvex and the equality E(V) ∩ 𝔻 = ⋃_{n=1}^{∞} intK̂_{n},

which can be easily obtained from the property K̂_{n} ∩ 𝔻 ⊂ intK̂_{n+1} proved above.

2) Let t be any real number and let e_{v} be the unit vector in the v-direction. Then for each n∈ℕ one has K_{n} + te_{v} ⊂K_{n+k} for all k big enough, therefore K̂_{n} + te_{v} ⊂K̂_{n+k} and consequently

E(V) + te_{v} = ⋃_{n}(K̂_{n} + te_{v})⊂⋃_{n} K̂_{n+k} = E(V).

3) It follows from strict pseudoconvexity of G×ℝ_{v} ⊂ℂ_{z,w}^{2} that the CRhull E(U ×ℝ_{v}) of the set U ×ℝ_{v} is single-sheeted, i.e. it is a subdomain of \overline{G} × ℝ_{v}. Since, by part 1, (E(U) ∩ G) × ℝ_{v} is pseudoconvex, we conclude that E(U × ℝ_{v}) ⊂ E(U × ℝ_{v}). Hence, we only need to prove the other inclusion E(U × ℝ_{v})⊂E(U × ℝ_{v}).

Let E_{1} ⊂ E_{2} ⊂ E_{3} ⊂ ⋯ be an exhaustion of U by compact subsets such that each set E_{n} is contained in the interior part of E_{n+1}. Consider also an exhaustion of U × ℝ_{v} by compact subsets K_{n} = E_{n} × [−n, n]. It follows from our assumption on simply-connectedness of the components of U, that for each n ∈ ℕ there exist finitely many disjoint smooth simply-connected domains D_{n}^{p} ⋐U such that E_{n} ⊂∪_{p}D_{n}^{p}. Consider piece-wise smooth 2-spheres

S_{n}^{p} = ((bD_{n}^{p}) × [−n − 1, n + 1]) ∪ (D_{n}^{p} × {−n − 1}) ∪ (D_{n}^{p} × {n + 1})

contained in bG, their smooth generic perturbations S̃_{n}^{p} ⊂bG × ℝ_{v} and given by Kruzhilin [Kr] 3-balls B_{n}^{p} with the boundary S̃_{n}^{p} foliated by holomorphic discs. Since bB_{n}^{p} = S̃_{n}^{p} ⊂ bG × ℝ_{v} ∖ K_{n}, we conclude for each p from pseudoconvexity of the domain G × ℝ_{v} ∖ K̂_{n} and from “Kontinuitätsatz” that B_{n}^{p}∩K̂_{n} = ∅ and that there is a neighbourhood V_{n}^{p} of B_{n}^{p} in G × ℝ_{v} such that V_{n}^{p} ⊂ E(U × ℝ_{v}). Denote by W_{n}^{p} the bounded component of the set G × ℝ_{v} ∖ B_{n}^{p}. Then, by strict pseudoconvexity of G × ℝ_{v}, there is a neighbourhood W_{n}^{p} of (bG × ℝ_{v}) ∩ \overline{W}_{n}^{p} in \overline{G} × ℝ_{v} which is contained in E(U × ℝ_{v}). Since V_{n}^{p} ⊂ E(U × ℝ_{v}) and W_{n}^{p} ⊂ E(U × ℝ_{v}), we conclude from Hartogs theorem on removability of compact singularities that W_{n}^{p} ⊂ E(U × ℝ_{v}) for all p. Then the inclusion K̂_{n} ⊂ ⋃_{p} W_{n}^{p} implies that K̂_{n} ⊂ E(U × ℝ_{v}) and hence E(U × ℝ_{v}) = ⋃_{n} K̂_{n} ⊂ E(U × ℝ_{v}). The proof of Theorem 1 is now completed. □

Proof of Theorem 2. We consider first the special case when the domain G is bounded. Then the set E(U) cannot have connected components with part of the boundary in two different components U_{α_{1}} and U_{α_{2}}. Indeed, let F be the connected component of bU_{α_{1}} such that U_{α_{1}} and U_{α_{2}} are contained in different connected components of the set bG ∖ F. Since F is connected, and since bG is diffeomorphic to a 2-sphere, the set bG ∖ F is the union of simply-connected domains {V_{β}}. We can assume that the domains V_{β} are enumerated in such a way that U_{α_{1}} ⊂ V_{1}. Let D^{1} ⊂ D^{2} ⊂ ⋯ be an exhaustion of the set bG \ F by open subsets such that for each k = 1, 2, ... one has D^{k} ⋐ D^{k+1}, the set D^{k} is the union of finitely many simply-connected

8 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINI

domains with smooth boundary and for each β the set D_{β}^{k} = D^{k} ∩ V_{β} consists of at most one connected component. Observe that according to our enumeration for the domains {D_{β}^{k}} one has U_{α_{1}} ⊂ ⋃_{k=1}^{+∞} D_{1}^{k}.

Let E_{1} ⊂E_{2} ⊂⋯ be an exhaustion of U by compact subsets such that E_{n} ⋐E^{o}_{n+1} and E_{n}∩U_{α} is connected for every n∈ℕ and every α. Let K̂_{n} be the A(G × ℝ_{v})-hull of K_{n} = E_{n}×[−n, n]. Since ⋃_{k} D^{k} = bG∖F ⊃U, it follows that for each n there is a number k(n) such that E_{n} ⊂D^{k(n)}. For each β such that D^{k(n)} ∩ V_{β} ≠ ∅ consider a piece-wise smooth 2-sphere

S_{β}^{k(n)} =((bD_{β}^{k(n)})×[−n−1, n+1])∪(D_{β}^{k(n)}×{−n−1})∪(D_{β}^{k(n)}×{n+1})

contained in bG. Smoothing the edges of S_{β}^{k(n)} and perturbing it a little bit we get a smooth generic 2-sphere S̃_{β}^{k(n)} ⊂ bG. It follows then from Kruz̆ilin [Kr] (see also Bedford-Klingenberg [BK]), that there is a 3-ball B_{β}^{k(n)} with the boundary S̃_{β}^{k(n)} foliated by holomorphic discs. Since, by our construction, the domains D_{β}^{k(n)} are disjoint, the 3-balls B_{β}^{k(n)} will also be disjoint. Since each ball B_{β}^{k(n)} is foliated by holomorphic discs with the boundary bS_{β}^{k(n)} ⊂bG ∖ K_{n}, we conclude from Lemma 1 and“Kontinuitätsatz” that B_{β}^{k(n)} ∩ K̂_{n} = ∅.

For each β denote by W_{β}^{k(n)} the bounded component of the set \overline{G} × ℝ_{v} ∖ B_{β}^{k(n)}. Then we have K̂_{n} ⊂ ⋃_{β} W_{β}^{k(n)}. Consider the sets W_{1} =

⋃_{n=1}^{+∞} W_{1}^{k(n)} and W_{2} = ⋃_{n=1}^{+∞} ⋃_{β≠1} W_{β}^{k(n)}. Since for each n the sets W_{β}^{k(n)} are

relatively open in \overline{G} × ℝ_{v} and disjoint, and since for each β the family of sets W_{β}^{k(n)} is increasing with respect to n, it follows that the sets W_{1} and W_{2} are disjoint and relatively open in \overline{G} × ℝ_{v}. The inclusion

K̂_{n} ⊂ ⋃_{β} W_{β}^{k(n)} for each n implies that E(U × ℝ_{v}) = ⋃_{n} K̂_{n} ⊂ W_{1} ∪ W_{2}.

Since U_{α_{1}} × ℝ_{v} ⊂W_{1} and U_{α_{2}} × ℝ_{v} ⊂(U ∖ U_{α_{1}}) × ℝ_{v} ⊂W_{2}, we conclude that the set E(U) cannot have connected components with parts of the boundary in two different connected components U_{α_{1}} and U_{α_{2}}.

In order to prove that E(U) = ⋃_{α} E(U_{α}) it is enough to show that for each connected component U_{α_{1}} of U one has E(U) ∩ W_{1} = E(U_{α_{1}}) (see the definition of W_{1} above). Since E(U)×ℝ_{v} = ⋃_{n} K̂_{n} and E(U_{α_{1}})×ℝ_{v} = ⋃_{n} K̂_{n}^{U_{α_{1}}} (here K_{n}^{U_{α_{1}}} = K_{n} ∩(U_{α_{1}} ×ℝ_{v})), it is sufficient to prove that for

each n we have that K̂_{n} ∩W_{1}^{k(n)} = K̂_{n}^{U_{α_{1}}}. It follows from Lemma 1 that for each n the set W_{1}^{k(n)} ∖(K̂_{n} ∩W_{1}^{k(n)}) is pseudoconvex and, therefore, K̂_{n} ∩ W_{1}^{k(n)} ⊂ K̂_{n}^{U_{α_{1}}}. The opposite inclusion K̂_{n}^{U_{α_{1}}} ⊂ K̂_{n} ∩ W_{1}^{k(n)} is obvious, since K̂_{n}^{U_{α_{1}}} ⊂ K̂_{n}. Hence K̂_{n} ∩ W_{1}^{k(n)} = K̂_{n}^{U_{α_{1}}} for each n and then

E(U) ∩ W_{1} = ⋃_{n}(K̂_{n} ∩ W_{1}^{k(n)}) = ⋃_{n} K̂_{n}^{U_{α_{1}}} = E(U_{α_{1}}).

The connectedness of the sets E(U_{α}) follows from the fact that E(U_{α})× ℝ_{v} = ⋃_{n} K̂_{n}^{U_{α}} and the connectedness of each set K̂_{n}^{U_{α}} which is due

to Shilov Idempotent Theorem (see, for example [G], p. 88). This completes the proof in the case of bounded domain G.

Consider now the general case when the domain G is unbounded. It follows from the argument of Proposition 3.4 in [ST] that there is an exhaustion of G by bounded domains G_{n} such that G_{n} × ℝ_{v} are piece-wise strictly pseudoconvex and G ∩ 𝔹_{n} ⊂ G_{n}, where 𝔹_{n} is the ball in ℂ_{z} × ℝ_{u} of radius n with centre at the origin. Moreover, since the domain G is assumed to be diffeomorphic to a 3-ball, the covering model G of G is simply-connected, hence, we can use in the mentioned above construction of G_{n} an exhaustion of G by simply-connected subdomains to insure that each domain G_{n} is also diffeomorphic to a 3-ball. Smoothing the edges of G_{n} (see e.g. [T]), we get an exhaustion of G by bounded domains G̃_{n} diffeomorphic to a 3-ball such that G̃_{n} × ℝ_{v} are strictly pseudoconvex and G∩𝔹_{n} ⊂ G̃_{n}. Denote by U_{n} the interior part of the set U ∩ bG̃_{n}. In view of the argument of Lemma 1, the set E(U_{n}) defined with respect to the domain G is the same as being defined with

respect to G̃_{n}. It is also easy to see that E(U) = ⋃_{n=1}^{+∞} E(U_{n}). Therefore,

applying the argument of the first part of the proof to G̃_{n} and U_{n} and passing to the limit as n → ∞ we obtain the required properties of E(U) in the general case. The proof of Theorem 2 is now completed. □

Remark. In the proof of Theorem 2 the use of Bedford-Klingenberg Theorem is similar to the arguments of Eroskin [Er].

4. Characterization of the CR-hull of rigid surfaces.

Let, as above, U be an open subset of bG, where G is a domain in ℂ_{z} × ℝ_{u} such that G × ℝ_{v} is strictly pseudoconvex. Since the CR-hull E(U × ℝ_{v}) of U × ℝ_{v} is equal to the union of CR-hulls of its connected components, and since for any exhaustion U_{1} ⋐ U_{2} ⋐ U_{3} ⋐ ⋯ of U one has E(U × ℝ_{v}) = ∪_{n=1}^{∞}E(U_{n} × ℝ_{v}), it is enough to give a description of the CR-hull of U × ℝ_{v} for U being a smooth bounded domain in bG which is in general position with respect to the canonical projection p : \overline{G} → \overline{G}. The last assumption means that all the selfintersections and intersections with bG of the curve p(bU) are transversal. In the rest of this section we will always assume that U ⊂bG is a smooth bounded domain in general position. In the case of an arbitrary domain U a characterization similar to the given below in Proposition 1 can be easily obtained by passing to the limit.

Let U^{−} = U ∩ bG^{−} and U^{+} = U ∩ bG^{+}, where bG^{−} and bG^{+} are the ”lower” and the ”upper” part of bG, respectively, defined in the Introduction. Denote by Σ the union of all connected components of the set p(U^{−}) ∩ p(U^{+}) whose closure has a nonempty intersection with bG. Further, denote U^{−} = p(U^{−}) ∖ Σ and U^{+} = p(U^{+}) ∖ Σ. Consider the function f^{−} defined on b U^{−} which equals h^{−} on the set b U^{−} ∖ bΣ and h^{+} on the set b U^{−} ∩ bΣ. Let F^{−} be the harmonic extension of f^{−} to U^{−}. Similarly, consider the function f^{+} defined on b U^{+} which equals h^{+} on the set b U^{+} ∖ bΣ and h^{−} on the set b U^{+} ∩ bΣ. Let F^{+} be the harmonic extension of f^{+} to U^{+}. Denote

Σ = {(z, u) : z = π(ζ), h^{−}(ζ) ≤ u ≤ h^{+}(ζ), ζ ∈ Σ},
U^{−} = {(z, u) : z = π(ζ), h^{−}(ζ) ≤ u < F^{−}(ζ), ζ ∈ U^{−}},
U^{+} = {(z, u) : z = π(ζ), F^{+}(ζ) < u ≤ h^{+}(ζ), ζ ∈ U^{+}},

and

U = Σ ∪ U^{−} ∪ U^{+}.

Now we observe that the sets Σ, U^{−} and U^{+} are disjoint and, therefore, U is a one-sheeted subdomain of \overline{G}. Indeed, the set Σ is disjoint from the sets U^{−} and U^{+}, since the corresponding set Σ is disjoint from the sets U^{−} and U^{+}. To show that the set U^{−} is disjoint from the set U^{+} we consider an arbitrary connected component ν of the set U^{−} ∩ U^{+}. Note that, by definition of the sets U^{−} and U^{+}, the set \overline{ν} is relatively compact in G. Denote by γ^{−} the set b U^{−}∩ \overline{ν} and by γ^{+} the set b U^{+}∩ \overline{ν}. Since the functions h^{−} and −h^{+} are strictly subharmonic, and since the functions F^{−} and F^{+} are harmonic, one has h^{−}(ζ) < F^{−}(ζ) < h^{+}(ζ) for all ζ ∈ U^{−} and h^{+}(ζ) < F^{+}(ζ) < h^{+}(ζ) for all ζ ∈ U^{+}. Hence, F^{−}(ζ) < h^{+}(ζ) = F^{+}(ζ) for ζ ∈ γ^{+} and F^{−}(ζ) = h^{−}(ζ) < F^{+}(ζ) for ζ ∈ γ^{−}. Since bν=γ^{−} ∪ γ^{+}, it follows that F^{−}(ζ) < F^{+}(ζ) for all ζ ∈ \overline{ν} which implies that the sets U^{−} and U^{+} are disjoint. Now we can formulate the main statement of this section.

Proposition 1. Let G be a domain in ℂ_{z} × ℝ_{u} such that G × ℝ_{v} is strictly pseudoconvex. Let U be an open bounded subdomain of bG such that bU is smooth and in general position with respect to the canonical projection p : \overline{G} → \overline{G}. Then for the CR-hull E(U × ℝ_{v}) of the set U × ℝ_{v} one has E(U × ℝ_{v}) = U × ℝ_{v}.

Proof. We show first that the CR-hull E(U × ℝ_{v}) of U × ℝ_{v} is singlesheeted, i.e. it is a subdomain of \overline{G} × ℝ_{v}. Let W_{1} ⋐ W_{2} ⋐ W_{3} ⋐ ⋯ be an exhaustion of U × ℝ_{v} by open subsets. For each n = 1, 2, … consider a bounded strictly pseudoconvex subdomain G_{n} of G × ℝ_{v} such that W_{n} ⊂ bG_{n}. It follows then from [St] that E(W_{n}) is equal to the complement in \overline{G}_{n} of the A(G_{n})-hull of bG_{n} ∖ W_{n}. In particular, E(W_{n}) is single-sheeted. Since E(U × ℝ_{v}) is equal to the union of the increasing sequence of hulls E(W_{n}), we conclude that E(U ×ℝ_{v}) is also single-sheeted.

Now we observe that the domain U × ℝ_{v} is pseudoconvex. Indeed, pseudoconvexity on the part (U ∩ bG) × ℝ_{v} of its boundary follows from strict pseudoconvexity of G × ℝ_{v}. For proving pseudoconvexity of U × ℝ_{v} inside G × ℝ_{v} we denote by I the set of those points of selfintersection of p(bU) which are contained in \overline{Σ}. Further, we denote by I the set

{(z, u) ∈ G : z = p(ζ), h^{−}(ζ) < u < h^{+}(ζ), ζ ∈ I}.

It is easy to see that the set I is a finite union of the vertical (i.e. parallel to the u-axis) open segments. The “interior” part (bU ∩ G) × ℝ_{v} of the boundary of U × ℝ_{v} is a Levi-flat surface constituted by pieces

{(z, w) ∈ G × ℝ_{v} : z = π(ζ), u = F^{−}(ζ), ζ ∈ U^{−}},
{(z, w) ∈ G × ℝ_{v} : z = π(ζ), u = F^{+}(ζ), ζ ∈ U^{+}},

and I×ℝ_{v} each of which is foliated by holomorphic curves. Levi-flatness of the first two sets follows directly from harmonicity of the functions F^{−} and F^{+}. The third set is just the finite union of holomorphic strips each of which is a segment in u-direction multiplied by ℝ_{v}. Since the surface (bU∩G)×ℝ_{v} is Levi-flat, we conclude that the domain U×ℝ_{v} is pseudoconvex on this part of its boundary and, therefore, the domain U × ℝ_{v} itself is pseudoconvex.

In the remaining part of the proof we use “Kontinuitätssatz” to show that U × ℝ_{v} ⊂E(U × ℝ_{v}). Since U × ℝ_{v} is proved to be pseudoconvex, it will imply that E(U × ℝ_{v}) = U × ℝ_{v}.

Note first that strict pseudoconvexity of the domain G × ℝ_{v} implies that there is a one-sided neighbourhood V × ℝ_{v} of U × ℝ_{v} filled by holomorphic discs with the boundary on U × ℝ_{v}. V × ℝ_{v} is obviously contained in E(U × ℝ_{v}). We proceed further in two steps. In the first step we “fill” the set Σ × ℝ_{v} by a continuous family of holomorphic discs. More precisely, for each point Q ∈ Σ we construct a family of holomorphic discs {D_{t}}, 0 ≤ t ≤ 1, such that D_{0} ⊂V×ℝ_{v}, bD_{t} ⊂V×ℝ_{v} for all t ∈ [0, 1] and D_{t} ∩ ({Q} × ℝ_{v}) ≠ ∅ for some t. Since the surface U ×ℝ_{v} is rigid, and since Q is an arbitrary point of Σ, this will imply by “Kontinuitätssatz” that the whole set Σ×ℝ_{v} is contained in E(U ×ℝ_{v}). In the second step we “fill” the rest of U×ℝ_{v} by a continuous family of holomorphic discs. Namely, for each point Q ∈ U^{−} ∪ U^{+} we construct holomorphic discs {D_{t}′}, 0 ≤ t ≤ 2, such that D′_{0} ⊂ V × ℝ_{v}, bD′_{t} ⊂ (Σ∪V)×ℝ_{v} for all t ∈ [0, 2] and D_{t}′∩({Q}×ℝ_{v}) ≠ ∅ for some t. Then, as above, by “Kontinuitätssatz” we conclude that E(U ×ℝ_{v}) = U×ℝ_{v} as it is stated in proposition.

Step 1. Let Q be a point of the set Σ. To simplify the proof we assume that the set Σ is connected. In the general case one can apply the argument below to each of the connected components of Σ. Consider a point ζ_{0} contained in the interior of one of the arcs constituting the set b G ∩ \overline{Σ}. Since the set Σ = p(Σ) is connected, there is a smooth curve γ(t) ⊂ Σ, 0 ≤ t ≤ 1, connecting the points ζ_{0} and p(Q) ∈ Σ and transversal to bG at the point ζ_{0}. For each T ≤ 1 denote by γ_{T} the part of γ corresponding to t ∈ [0, T]. Let W be a neighbourhood of ζ_{0} in \overline{G} so small that p^{−1}(W)⊂V. Choose δ > 0 small enough so that for the δ-neighbourhood V_{δ}(γ) of γ in \overline{G} one has V_{δ}(γ) ∩ bΣ⊂W. Consider T_{0} < 1 such that V_{δ}(γ_{T})⊂W for all T ∈ [0, T_{0}]. For δ_{1} ∈ (0, δ) and for each T ∈ [T_{0}, 1] consider the domain A_{T} = V_{δ}(γ_{T}) ∖ \overline{V}_{δ_{1}}(γ_{T})⊂\overline{G}. The boundaries of the domains A_{T} are smooth everywhere except for some points of the set bΣ ∩ W where they have transversal intersection with bG. It is easy to see that the domains A_{T} depend continuously on T. For each T ∈ [T_{0}, 1] define the function ϕ_{T} (ζ) on bA_{T} to be equal to h^{−}(ζ) on bV_{δ}(γ_{T}) ∩ G, to be equal to h^{+}(ζ) on bV_{δ_{1}}(γ_{T}) ∩ G and finally to be equal to h^{−}(ζ) = h^{+}(ζ) on bA_{T} ∩ bG. Then {ϕ_{T} } is a family of functions of Hölder class C^{1/2}(bA_{T}) depending continuously on T (each of these functions will behave like ±\sqrt{dist(ζ, bG)} as ζ → bG).

Let F_{T} be the harmonic extension of ϕ_{T} to A_{T}. Since each domain A_{T} is constructed to be simply-connected, for each T ∈ [T_{0}, 1] there exists a holomorphic function G_{T} defined on A_{T} and continuous \overline{A}_{T} such that Re G_{T} = F_{T} (these functions will actually be also in some Hölder class on Ā_{T}). One can obviously choose the functions G_{T} to be continuously dependent of T. Consider the family of holomorphic discs

D_{T} = {(z, w) : z = π(ζ), w = G_{T} (ζ), ζ ∈ Ā_{T} }, T_{0} ≤ T ≤ 1.

Then, by our construction, D_{T} depend continuously on T, for each T one has bD_{T} ⊂ U × ℝ_{v} and, since A_{T_{0}} ⊂ W, we conclude that D_{T_{0}} ⊂ V × ℝ_{v}. Moreover, the family of domains A_{T} and the functions F_{T} were constructed in such a way that the graph of some function F_{T} in our family have to contain the point Q and, hence, the corresponding holomorphic disc D_{T} will contain a point of {Q} × ℝ_{v}. It follows then from “Kontinuitätssatz” that this point of {Q} × ℝ_{v} is contained in E(U × ℝ_{v}). Since the surface U × ℝ_{v} is rigid, the whole line {Q} × ℝ_{v} is contained in E(U × ℝ_{v}), and since Q is an arbitrary point of Σ, we finally conclude that Σ × ℝ_{v} ⊂E(U × ℝ_{v}).

Step 2. Here we need to prove that U^{−} ∪ U^{+} ⊂ E(U × ℝ_{v}). Since the argument for both sets U^{−} and U^{+} is the same, we will show that U^{−} ⊂ E(U × ℝ_{v}). We can assume in what follows that the set U^{−} is connected. If not, we use the same argument for each connected component of U^{−} separately. In particular, our assumption implies that U^{−} is a (connected) domain in \overline{G}.

Let Q be a point of U^{−}. Then Q = (π(ζ_{Q}), u_{Q}) for some ζ_{Q} ∈ U^{−}. It follows from the definition of U^{−} that ε =^{def} F^{−}(ζ_{Q}) − u_{Q} > 0, where F^{−} is the function defining the ”upper” (with respect to the u-direction) part of b U^{−}. Remind, that by definition of the function f^{−} = F^{−}|_{b U^{−}} it is continuous everywhere except maybe for finitely many points contained in b U^{−}∩ b U^{+}.

Now we consider a new domain Ũ^{−} obtained by shrinking the domain U^{−} on the part b U^{−} ∖ bΣ of its boundary and enlarging it on the resting part b U^{−} ∩ bΣ of the boundary so that the domain Ũ^{−} will have a smooth boundary. Further, we approximate the function f^{−} by a smooth function f̃^{−} defined on b Ũ^{−}. More precisely, let Ũ^{−} be a smoothly bounded domain over G with projection π̃ : Ũ^{−} → G and let f̃^{−} be a smooth function on b Ũ^{−} with harmonic extension F̃^{−} to Ũ^{−} such that π̃(Ũ^{−})⊂ U^{−} ∪ Σ,

Γ(f̃^{−}) =^{def} {(z, u) : z = π(π̃(ξ)), u = f̃^{−}(ξ), ξ ∈ b Ũ^{−}}⊂U^{−} ∪ Σ

and F̃^{−}(ξ_{Q}) > F^{−}(ζ_{Q}) − ε/2, where ξ_{Q} = π̃^{−1}(ζ_{Q}). The last inequality implies that u_{Q} < F̃^{−}(ξ_{Q}) − ε/2. The problem now is that the domain Ũ^{−} might be multi-connected, while for being able to use “Kontinuitätssatz” we need to have a simply-connected domain. That is why we need to modify the domain Ũ^{−} and the function f̃^{−} further. To do this we use the argument of Lemma 5.1 in [ST]. For the convenience of reading we remind the construction used there.

Let Γ_{1}, Γ_{2}, …, Γ_{m} be the connected components of b Ũ^{−}. Consider the universal covering ˜ π̃ : Δ → Ũ^{−} of Ũ^{−} by the unit disc Δ. Let γ_{1}, γ_{2}, …, γ_{m−1} be a disjoint family of smooth curves in Ũ^{−} such that each curve γ_{i}, i = 1, …, m − 1, connects the corresponding curves Γ_{i} and Γ_{i+1} and non of the curves γ_{i} contains the point ξ_{Q}. Since, by the definition of U^{−} and Σ, non of the connected components of b U^{−} is contained in Σ, we can choose the curves γ_{i} to be contained in π̃^{−1}(U^{−}). Then the domain Ũ^{−} \ ⋃_{i=1}^{m−1} γ_{i} is simply-connected. Hence, the set ˜ π̃^{−1}(Ũ^{−} \ ⋃_{i=1}^{m−1} γ_{i}) is the disjoint union of a countable family of simply-connected subdomains {F_{σ}} of Δ. For each σ denote by F_{σ}^{∗} the union of the set \overline{F}_{σ} ∩ Δ with all the domains F_{σ′} such that \overline{F}_{σ} ∩ \overline{F}_{σ′} ≠ ∅.

Choose now one of the domains {F_{σ}} and denote it by F_{0}. Denote by F_{1} the corresponding domain F_{0}^{∗}. Define inductively for each n = 2, 3, … a domain F_{n} as the union of all the domains F_{σ}^{∗} such that F_{σ} ⊂ F_{n−1}. It follows directly from the definition that F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ and that ⋃_{n=0}^{+∞} F_{n} = Δ.

The following lemma was proved in [ST] (Lemma 5.1).

Lemma. There exists a positive number θ < 1 such that the inequality for harmonic measure

ω(η, bF_{n} ∩ Δ, F_{n}) ≤ θ^{n}

holds true for all η ∈ \overline{F}_{0} and all n = 1, 2, . . ..

Denote by η_{Q} the point ˜ π̃^{−1}(ξ_{Q})∩F_{0}. Let ˜ f̃_{n}^{−} be the function on bF_{n} such that ˜ f̃_{n}^{−}(η) = f̃^{−}(˜π̃(η)) for η ∈ bF_{n}∩bΔ and ˜ f̃_{n}^{−}(η) = h^{−}(π̃(˜π̃(η))) for η ∈ bF_{n} ∩ Δ. Consider the harmonic extension ˜ F̃_{n}^{−} of the function ˜f̃_{n}^{−} to the domain F_{n}. It follows then from Lemma 5.1 in [ST] recalled above that ˜ F̃_{n}^{−} > F̃^{−}(ξ_{Q}) − ε/2 if n is big enough. In particular, u_{Q} < ˜ F̃_{n}^{−}(η_{Q}). Note, that by the construction of F_{n} and ˜ F̃_{n}^{−} one has π̃(˜π̃(F_{n}))⊂ U^{−} ∪ Σ and

Γ( ˜F̃_{n}^{−}) =^{def} {(z, u) : z = π(π̃(˜π̃(η))), u = ˜ f̃_{n}^{−}(η), η ∈ bF_{n}}⊂U^{−} ∪ Σ.

Now we are in position to construct the family of holomorphic discs required by “Kontinuitätssatz”. Define for t ∈ [1, 2] the function ϕ_{t}′ on bF_{n} as ϕ_{t}′(η) = (1 − t) ˜f̃_{n}^{−} + (2 − t)h^{−}(π̃(˜π̃(η))) and for each t consider the harmonic extension ˜ F̃_{t} of the function ϕ_{t}′ to the domain F_{n}. Since F_{n} is a simply-connected domain with a piece-wise smooth boundary, and since the functions ϕ_{t}′ are smooth on bF_{n}, it follows that there is a one-parameter family of holomorphic functions G_{t}′ defined F_{n} and continuous on \overline{F}_{n} which depend continuously on the parameter t and such that Re G_{t}′ = ˜ F̃_{t}. Consider the family of holomorphic discs

D_{t}′ = \{(z, w) : z = π(π̃(˜π̃(η))), w = G_{t}′(η), η ∈ \overline{F}_{n}\}, 1 ≤ t ≤ 2.

Then, by our construction of F_{n} and ϕ_{t}′, for each t ∈ [1, 2] one has bD_{t}′ ⊂ (U^{−} ∪ Σ) × ℝ_{v} and, since ˜ F̃_{1}(η) = h^{−}(π̃(˜π̃(η))) for all η ∈ bF_{n}, one also has bD_{1}′ ⊂U × ℝ_{v}.

To finish the proof we use the fact that the domain F_{n} is simplyconnected. Namely, consider a one-parameter family of closed smoothly bounded simply-connected subdomains A_{t}′, 0 ≤ t ≤ 1, of F_{n} which depends continuously on t and such that A_{1}′ = F_{n} and A_{0}′ is a point in F_{n}. For each t ∈ [0, 1] let ˜ F̃_{t} be the harmonic extension of the function ϕ_{t}′(η) = h^{−}(π̃(˜π̃(η))) from bA_{t}′ to A_{t}′. Consider a one-parameter family of holomorphic functions G_{t}′ defined on the respective domains A_{t}′ which depend continuosly on t and such that Re G_{t}′ = ˜ F̃_{t}. Then for the family of holomorphic discs

D_{t}′ = {(z, w) : z = π(π̃(˜π̃(η))), w = G′_{t}(η), η ∈ Ā_{n}},

0 ≤ t ≤ 1, one obviously has that D_{0}′ is just a point in U × ℝ_{v} and, for each t ∈ [0, 1], bD_{t}′ ⊂ U × ℝ_{v}. If we consider the whole family of discs D_{t}′, 0 ≤ t ≤ 2, then it will depend continuously on t, D_{0} will be a point in U × ℝ_{v} and for each t one will have bD_{t}′ ⊂ (U^{−} ∪ Σ) × ℝ_{v}. Observe now that by our construction one has

⋃_{0≤t≤2}π_{z,u}(D_{t}′)={(z, u): π(π̃(˜π̃(η))), h^{−}(π̃(˜π̃(η))) ≤ u ≤ ˜ F̃_{n}^{−}(η), η ∈ F_{n}},

where π_{z,u} : ℂ_{z,w}^{2} → ℂ_{z} × ℝ_{u} is the projection. Since u_{Q} < ˜ F̃_{n}^{−}(η_{Q}), it follows that there is a disc D_{t_{0}}′ in our family such that Q ∈ π_{z,u}(D_{t_{0}}′). This means that there is a point of {Q} × ℝ_{v} contained in D_{t_{0}}′. It follows then from “Kontinuitätssatz” that this point is contained in the CR-hull of the set (U^{−} ∪ Σ) × ℝ_{v}, which is, in view of Step 1, contained in E(U × ℝ_{v}). Rigidity of the domain E(U × ℝ_{v}) imply now that the line {Q} × ℝ_{v} is contained in E(U × ℝ_{v}) and then, by the choice of Q, the whole set U^{−} ∪ U^{+} is contained in E(U × ℝ_{v}). This complete the argument of Step 2 and the proof of Proposition 1. □

5. Proof of Theorems 3 and 4.

Proof of Theorem 3. The fact that the sets E(V_{φ}^{±}) are open in \overline{G}×ℝ_{v} follows from the part 1 of Theorem 1 and the fact that these sets can be represented in the form (1) and (2) follows from the same argument as in the part 2 of Theorem 1. The property Φ^{±}|_{U} = φ follows directly from the definition of E(V_{φ}^{±}). □

Proof of Theorem 4. We start with the proof of continuity of the functions Φ^{±} at the points of the set U. Since the domain G × ℝ_{v} is strictly pseudoconvex, there are one-sided neighbourhoods V^{+} and V^{−} of the sets V_{φ}^{+} and V_{φ}^{−} in \overline{G} × ℝ_{v}, respectively, filled by holomorphic discs with the boundary on the respective set V_{φ}^{+} and V_{φ}^{−}. Since V^{+} ⊂ E(V_{φ}^{+}) and V^{−} ⊂ E(V_{φ}^{−}), and since directly from the definition of Φ^{±} one has Φ^{−}(z, u) ≤ Φ^{+}(z, u) for (z, u)∈E(U), we conclude that Φ^{±} are continuous at the points of U.

To prove continuity of the functions Φ^{±} in the interior points of E(U) we need the following

Lemma 2. There is an increasing sequence of open relatively compact subsets {E_{n}} of E(U) such that

1) E_{n} × ℝ_{v} is piece-wise strictly pseudoconvex for every n ∈ ℕ

and

2) For each compact subset K of E(U) one has K ⊂ \overline{E}_{n} for all n big enough.

Proof. Let, as above, E_{1} ⊂ E_{2} ⊂ E_{3} ⊂ ⋯ be an exhaustion of U by compact subsets and let K_{n} = E × [−n, n]. Since E(U) = ⋃_{n} π_{z,u}(K̂_{n}),

where π_{z,u} : ℂ_{z,w}^{2} → ℂ_{z} × ℝ_{u} is the projection, it is enough to construct for each n ∈ ℕ an open subset E_{n} of E(U) satisfying property 1 and such that π_{z,u}(K̂_{n}) ⊂ \overline{E}_{n}. For n being fixed, we can assume in the construction of E_{n} that the domain G is bounded. If not, we use the same argument as in Lemma 1. Namely, for r large enough consider the connected component G_{r} of G∩𝔹_{r} containing E_{n} on the boundary. Smoothing the edges of G_{r} (see e.g. [T]), we obtain a bounded domain G̃_{r} such that G̃_{r} × ℝ_{v} is strictly pseudoconvex. Finally, we can replace the domain G by G̃_{r} and the set U by the interior part of the set U ∩ bG̃_{r}.

Let ϱ be the defining function of the domain G, i.e. G = {(z, u) : ϱ(z, u) < 0} and grad ϱ ≠ 0 on bG. For each δ > 0 consider the domain G^{δ} = {(z, u) : ϱ(z, u) < δ} and the set

U_{n}^{δ} = {(z, u) : ϱ(z, u) = δ, dist ((z, u), E_{n}) < ε_{n}/2},

where ε_{n} = dist(E_{n}, bU) and E_{n} is the corresponding compact from the exhaustion of U. Let E(U_{n}^{δ}) be the set defined in the same way as the set E(U) with G and U replaced by G^{δ} and U_{n}^{δ}, respectively. Since for all δ small enough the domains G^{δ}×ℝ_{v} are strictly pseudoconvex, there are uniformly big one-sided neighbourhoods V_{n}^{δ} of the sets U_{n}^{δ} × ℝ_{v} in the respective domains \overline{G}^{δ} × ℝ_{v} filled by holomorphic discs with the boundary on the corresponding set U_{n}^{δ} ×ℝ_{v}. Then, for some δ_{0} > 0 one has K_{n} ⊂ V_{n}^{δ_{0}} ⊂ E(U_{n}^{δ_{0}}) × ℝ_{v}. Therefore, by the definition of E(U_{n}^{δ_{0}}), one also has that K̂_{n} ⊂E(U_{n}^{δ_{0}}) × ℝ_{v}, i.e. π_{z,u}(K̂_{n})⊂E(U_{n}^{δ_{0}}).

It also follows from strict pseudoconvexity of the domains G^{δ} × ℝ_{v} that there are uniformly big one-sided neighbourhoods Ṽ _{n}^{δ} of the sets (bG^{δ}∖\overline{U}_{n}^{δ})×ℝ_{v} in the respective domains \overline{G}^{δ}×ℝ_{v} filled by holomorphic discs with the boundary on the corresponding set (bG^{δ} ∖ \overline{U}_{n}^{δ}) × ℝ_{v}. Since, by definition, the set E(U_{n}^{δ}) × ℝ_{v} is the union of the hulls of compacts exhausting the set U_{n}^{δ} ×ℝ_{v}, and since, by Lemma 1, the discs filling Ṽ _{n}^{δ} cannot intersect these hulls, it follows that bG ∩ E(U_{n}^{δ}) ⊂ U and, therefore, E(U_{n}^{δ})∩\overline{G}⊂E(U) for all δ small enough. We can assume that this inclusion holds for δ_{0} chosen above (otherwise we choose the smallest of these two δ’s). In view of Theorem 1, the interior part of the set E(U_{n}^{δ_{0}}) × ℝ_{v} is pseudoconvex, hence the function

h(z, u) = − log dist((z, u), bE(U_{n}^{δ_{0}}) × ℝ_{v})

is plurisubharmonic in the interior part of the set E(U_{n}^{δ_{0}})×ℝ_{v}. Smoothing this function and adding the function θ(|z|^{2}+u^{2}), θ > 0 being small enough, we get a strictly plurisubharmonic function h̃(z, u). Choose η > 0 sufficiently big such that the domain E_{η}(U_{n}^{δ_{0}}) = {(z, u) : h̃(z, u) < η} is close enough to E(U_{n}^{δ_{0}}) and bE_{η}(U_{n}^{δ_{0}}) is transversal to bG. Define E_{n} = E_{η}(U_{n}^{δ_{0}}) ∩ G. It follows then from the construction of E_{η}(U_{n}^{δ_{0}}) that the domain E_{n} × ℝ_{v} is piece-wise strictly pseudoconvex and that π_{z,u}(K̂_{n})⊂\overline{E}_{n}. This completes the proof of the lemma. □

Now we can proceed the same way as in [ST]. Namely, for each n ∈ ℕ consider the subset A_{n} = bG ∩ \overline{E}_{n} of bE_{n}. Further, for each s > 0 define the subset B_{n,s} of bE_{n}:

B_{n,s} = {P ∈ bE_{n} : dist (P, bG) ≥ 1/s}.

Consider a one-parameter family of continuous functions φ_{n,t}, t ∈ ℝ, defined on bE_{n}, continuously depending on the parameter t such that: φ_{n,t} = φ on A_{n}, where φ is the given continuous function on U, φ_{n,t} = t on B_{n,|t|} and φ_{n,t_{1}} ≤ φ_{n,t_{2}} if t_{1} ≤ t_{2}. Then, in view of the construction of E_{n}, we can apply Theorem 2 of [CS] to the functions φ_{n,t} and obtain for each t a function Φ_{n,t} ∈ C(\overline{E}_{n}) such that Φ_{n,t} = φ_{n,t} on bE_{n} and the graph Γ(Φ_{n,t}) ∖ Γ(φ_{n,t}) is (locally) foliated by 1-dimensional complex submanifolds. Define functions Φ̃^{±} on E(U) as

Φ̃^{+}(P) = lim_{n→+∞} lim_{t→+∞} Φ_{n,t}(P) and Φ̃^{−}(P) = lim_{n→+∞} lim_{t→−∞} Φ_{n,t}(P).

Remark. We will prove later (see Lemma 5) that the functions Φ̃^{±} coincide with Φ^{±}, respectively, and then we only use the notation Φ^{±}.

In the next lemma we prove the uniform boundedness of the functions Φ_{n,t} on compact subsets of E(U).

Lemma 3. For each compact subset K of E(U) there exist a constant C > 0 and a number N such that |Φ_{n,t}(P)| ≤ C for all P ∈ K, t ∈ ℝ and n ≥ N.

Proof. Let K be a compact subset of E(U) and let K_{1}^{±} ⊂K_{2}^{±} ⊂K_{3}^{±} ⊂⋯ be exhaustions by compact subsets of the sets V_{φ}^{±}, respectively, such that for every n ∈ ℕ each set K_{n}^{±} is a subset of the interior part of the respective set K_{n+1}^{±}. Since, by Theorem 3, π_{z,u}(E(V_{φ}^{±})) = E(U), it follows that there is a number L such that K ⊂ π_{z,u}(K̂_{L}^{±}). Consider a number N such that π_{z,u}(K̂_{L}^{±}) ⊂ E_{N}. Then for each n ≥ N and t ∈ ℝ the graph Γ(Φ_{n,t}) of the function Φ_{n,t} does not intersect the sets K̂_{L}^{±}.

Since the argument is the same for both sets K̂_{L}^{±}, we prove this property for the set K̂_{L}^{+}.

Indeed, for n ≥ N and t ∈ ℝ being fixed, it follows from continuity of Φ_{n,t} on E_{n} and compactness of K̂_{L}^{+} that Γ(Φ_{n,t} − C) ∩ K̂_{L}^{+} = ∅ for C > 0 big enough. Since, by definition of φ_{n,t} and the choice of E_{N}, Γ(φ_{n,t} − s) ∩ K̂_{L}^{+} = ∅ for s ≥ 0, and since, by Lemma 1, the domain G × ℝ_{v} ∖ K̂_{L}^{+} is pseudoconvex, we conclude from Levi-flatness of Γ(Φ_{n,t} − s), 0 ≤ s ≤ C, that Γ(Φ_{n,t}) ∩ K̂_{L}^{+} = ∅.

To complete the proof of Lemma 3 we observe that the properties Γ(Φ_{n,t}) ∩ K̂_{L}^{±} = ∅ and K ⊂ π_{z,u}(K̂_{L}^{±}) imply that |Φ_{n,t}(P)| ≤ C for all P ∈ K, t ∈ ℝ and n ≥ N, where C = max{|v| : (z, u + iv) ∈ K_{L}^{±}}. □

To show that the graphs Γ(Φ̃^{±}) ∖ Γ(φ) are foliated by holomorphic curves we follow the argument of section 6 in [ST]. Namely, from Lemma 3 we conclude, using Lemmas 3.2-3.5 in [Sh], that for each point P = (z_{0}, u_{0}) ∈ E(U)∩G there exists R^{∗} > 0 (which depends only on dist (P, bE(U)) and the constant C given by Lemma 3 for compact K being a fixed neighbourhood of P) such that for each t ∈ ℝ and all n big enough there is a holomorphic function

f_{n,t}^{P}(z) = u_{n,t}^{P}(z) + iv_{n,t}^{P}(z) : Δ_{R^{∗}}(z_{0}) → ℂ_{w}

with the properties u_{n,t}^{P}(z_{0}) = u_{0} and Γ(f_{n,t}^{P})⊂Γ(Φ_{n,t}), where Δ_{R^{∗}}(z_{0}) = {|z − z_{0}| < R^{∗}}. Then, in view of Montel’s theorem, we can pass to the limit as t → +∞ and n → +∞ (for details see the proof of Lemma 6.2 in [ST]) and obtain a holomorphic function

f^{P} (z) = u^{P} (z) + iv^{P} (z) : Δ_{R^{∗}}(z_{0}) → ℂ_{w}

such that f^{P} (z_{0}) = u_{0} and Γ(f^{P}) ⊂ Γ(Φ̃^{+}). Similarly, as t → −∞ and n → +∞ we get the corresponding holomorphic disc in Γ(Φ̃^{−}). Finally, using the argument of Proposition 6.3 in [ST], we obtain foliations of the whole graphs Γ(Φ̃^{±}) ∖ Γ(φ) by holomorphic curves constituted by mentioned above holomorphic discs.

For the leaves of these foliations we have the following

Lemma 4. Each holomorphic leaf contained in one of the sets Γ(Φ̃^{±})∖ Γ(φ) has an accumulation point in Γ(φ).

Proof. We argue by contradiction assuming that there is a holomorphic leaf L contained in one of the sets Γ(Φ̃^{±}) ∖ Γ(φ) such that L has no accumulation point in Γ(φ). Since both cases can be treated in the same way, we can assume that L ⊂ Γ(Φ̃^{−}) ∖ Γ(φ). The inclusion π_{z,u}(L) ⊂ E(U) implies that π_{z,u}(L) ∩ π_{z,u}(K̂_{n}) ≠ ∅ for some n ∈ ℕ. Let V be a relatively compact neighbourhood of the set π_{z,u}(K̂_{n}) in E(U) such that dist (bE(U) ∩ G, V) > 0. It follows then from our assumption and from continuity of Φ̃^{±} near U, that π_{z,u}(L) ∩ V ⋐ G. Since, by Lemma 3, Φ̃^{−} is bounded over compact subsets of E(U), we have from the construction of holomorphic leaves in Γ(Φ̃^{−}) that the closure \overline{L} of the leaf L in \overline{V} × ℝ_{v} is compact and that the set \overline{L} ∩ (V × ℝ_{v}) is foliated by holomorphic curves. Therefore, the domain V × ℝ_{v} ∖ \overline{L} is pseudoconvex near \overline{L}. Then, considering the function ϱ(P) = − log dist(P, \overline{L}) + ε(|z|^{2} + |w|^{2}) with small enough ε > 0 and smoothing it, if necessary, we construct (as a superlevel set of the constructed function) a strictly pseudoconcave neighbourhood V of \overline{L} in V′ × ℝ_{v} (we mean here the strict pseudoconcavity of V near the part bV ∩ (V′ × ℝ_{v}) of its boundary) such that V ⋐ G × ℝ_{v}, where V′ is a slightly smaller than V neighbourhood of π_{z,u}(K̂_{n}) in E(U). Since the set \overline{V} is compact, there is a big enough constant C such that (\overline{V} +Ce_{v})∩K̂_{n} = ∅. Then the property π_{z,u}(L)∩π_{z,u}(K̂_{n}) ≠ ∅ implies that for some t ≥ 0 the set \overline{V} +(C −t)e_{v} will touch for the first moment the set K̂_{n}. This contradicts pseudoconvexity of G × ℝ_{v} \ K̂_{n} given by Lemma 1. □

Continuity of the functions Φ̃^{±} in the interior points of E(U) follows now directly from the argument of Lemma 7.2 in [ST] with Lemma 7.1 replaced by just proved Lemma 4. Hence, we have proved that the functions Φ̃^{±} define continuous Levi-flat extensions of the given function φ ∈ C(U) to the domain E(U).

Lemma 5. The functions Φ^{±} defining the sets W^{±} coincide with the respective functions Φ̃^{±}.

Proof. Since the argument is the same for both functions, we prove the statement for the function Φ^{+}. First, we will show that Φ^{+} ≤ Φ̃^{+} in E(U). Consider for each ε > 0 and m ∈ ℕ the graph Γ(φ_{m,t} + ε) of the function φ_{m,t} + ε. It follows then from the definitions of V_{φ}^{+}, φ_{m,t} and Theorem 3 that for m, ε and t (big enough) being fixed one has Γ(φ_{m,t} + ε) ⊂ K̂_{n}^{+} if n is sufficiently big. Therefore, by the maximum principle, we conclude from the Levi-flatness of Γ(Φ_{m,t} + ε) that Γ(Φ_{m,t} + ε) ⊂ K̂_{n}^{+}. This implies, by the definition of Φ̃^{+} and E(V_{φ}^{+}), that Γ(Φ̃^{+} + ε) ⊂ \overline{E(V_{φ}^{+})}. Hence, Φ^{+} ≤ Φ̃^{+} + ε for each ε > 0 and, therefore, Φ^{+} ≤ Φ̃^{+}.

To prove the other inequality Φ̃^{+} ≤ Φ^{+} we argue by contradiction and assume that Φ^{+}(P_{0}) < Φ̃^{+}(P_{0}) for some point P_{0} ∈ E(U). It follows then from the definition of Φ^{+} and E(V_{φ}^{+}) that there exist a number n_{0} ∈ ℕ and a point (P_{0}, v_{0}) ∈ K̂_{n_{0}}^{+} such that v_{0} < Φ̃^{+}(P_{0}). Therefore, by the definition of Φ̃^{+}, there are m_{0} ∈ ℕ and t_{0} ∈ ℝ such that v_{0} < Φ_{m_{0},t_{0}}(P_{0}). We can also assume that m_{0} is chosen so big that the domain E_{m_{0}} (where the function Φ_{m_{0},t_{0}} is defined) has the property dist (bE_{m_{0}} ∩ G, π_{z,u}(K̂_{n_{0}}^{+})) > 0. Since Γ(Φ_{m_{0},t_{0}} − C) does not intersect

K̂_{n_{0}}^{+} for C big enough, and since, by the choice of m_{0}, Γ(φ_{m_{0},t_{0}} − s) does not intersect K̂_{n_{0}}^{+} for all s ≥ 0, we conclude from pseudoconvexity of the domain G × ℝ_{v} ∖ K̂_{n_{0}}^{+} (see Lemma 1) and from Levi-flatness of Γ(Φ_{m_{0},t_{0}} − s), 0 ≤ s ≤ C, that Γ(Φ_{m_{0},t_{0}} − s) ∩ K̂_{n_{0}}^{+} = ∅ for all 0 ≤ s ≤ C. This contradicts the inequality v_{0} < Φ_{m_{0},t_{0}}(P_{0}) and proves Lemma 5. □

The last statement of Theorem 4, saying that for each function Φ ∈ C(E(U)) such that Φ|_{U} = φ and Γ(Φ) ∖ Γ(φ) is Levi-flat one has Φ^{−}(z, u) ≤ Φ(z, u) ≤ Φ^{+}(z, u) for all (z, u) ∈ E(U), follows from pseudoconvexity of the domains G × ℝ_{v} ∖ (K̂_{n}^{−} ∪ K̂_{n}^{+}) by the same argument as in the proof of Lemma 3. The proof of Theorem 4 is now completed. □

Remark. Denote U_{c} = bG \ \overline{U} and let E^{∗}(U) be the union of all connected components of the set E(U ∪ U_{c}) which have some points of U on the boundary. Consider a function ψ ∈ C(U_{c}). Applying Theorem 4 to the set U ∪U_{c} instead of U and to the function on U ∪U_{c} which is equal to φ on U and to ψ on U_{c}, we obtain Levi-flat extensions of φ to the set E^{∗}(U) which is in general bigger than E(U) (see Example 1 in Section 7). If the domain G is diffeomorphic to a 3-ball, then, in view of Theorem 2, E^{∗}(U) = E(U) and these Levi-flat extensions of φ do not depend on the choice of ψ, being equal to the functions Φ^{±} given by Theorem 4 for the set U and the function φ.

6. On maximality of E(U).

Proof of Theorem 5. We can assume, without loss of generality, that the set U is connected. If not, we can apply the argument below to all the connected components of U. Since both cases U ⊂ bG^{−} and U ⊂bG^{+} can be treated in the same way, in what follows we assume that U ⊂ bG^{−}. Consider an exhaustion U_{1} ⋐ U_{2} ⋐ U_{3} ⋐ ⋯ of U by domains with smooth boundary and for each n = 1, 2, … let U_{n} = p(U_{n}), where p : \overline{G} → \overline{G} is the canonical projection. Further, consider U′ = p(U′), where U′ ⊂bG^{−} is the simply-connected domain given by assumption of the theorem. Denote by A_{n} the set consisting of all compact connected components of U′ ∖ U_{n}. Then the sets U_{n}′ = U_{n} ∪ A_{n} are simplyconnected subdomains of U′ such that U_{1}′ ⋐ U_{2}′ ⋐ U_{3}′ ⋐ ⋯ ⊂ U′ and U′ = ∪_{n=1}^{+∞}U_{n}′. For each n = 1, 2, … define the function H_{n}′ on the domain U′ to be equal to h^{−} on the set U′ ∖ A_{n} and to be the harmonic extension of h^{−} from bA_{n} to A_{n} on the set A_{n}. Further, define the function H_{n}′′ on the domain U′ to be equal to h^{−} on the set U′ ∖ U_{n}′ and to be the harmonic extension of h^{−} from b U_{n}′ to U_{n}′ on the set U_{n}′. Then all the defined functions are subharmonic on U′ and ⋯ ≤ H_{3}′ ≤ H_{2}′ ≤ H_{1}′ ≤ H_{1}′′ ≤ H_{2}′′ ≤ H_{3}′′ ≤ · · ·. Let H′(ζ) = lim_{n→+∞} H_{n}′(ζ) and H′′(ζ) = lim_{n→+∞} H_{n}′′(ζ), ζ ∈ U′. Define the

subdomain U of \overline{G} by

U = {(z, u) : z = π(ζ), H′(ζ) < u < H′′(ζ), ζ ∈ U′} ∪ U.

Now we observe that U ⊂ E(U). Indeed, let n be fixed and for each connected component Γ_{n}^{α} of the set b U_{n} consider the harmonic extension F_{n}^{α} of the function h^{−} from Γ_{n}^{α} to the domain B_{n}^{α} bounded by Γ_{n}^{α} in G. Note, that one of these domains is U_{n}′ (it corresponds to Γ_{n}^{α} = b U_{n}′) and the others are just the connected components of the set A_{n}. Since each of the domains B_{n}^{α} is simply-connected, and since the function h^{−} is smooth in G, for each α there is a function G_{n}^{α} holomorphic in B_{n}^{α} and smooth on \overline{B}_{n}^{α} such that Re G_{n}^{α} = F_{n}^{α}. For each α and each t ∈ ℝ consider the disc

D_{n,t}^{α} = {(z, w) : z = π(ζ), w = G_{n}^{α}(ζ) + it, ζ ∈ B_{n}^{α}}

and then define the set M_{n} to be the union of the set U_{n} × {0} with the boundaries of all discs D_{n,t}^{α} such that \overline{D}_{n,t}^{α} ∩ \{v = 0\} ≠ ∅. Since M_{n} is a compact subset of U × ℝ_{v}, it follows that M̂_{n} ⊂ E(U × ℝ_{v}). Consider the set

U_{n} = {(z, u) : z = π(ζ), H′_{n}(ζ) < u < H′′_{n}(ζ), ζ ∈ U′_{n}}.

Since the set U_{n} × {0} ⊂ ℂ_{z,w}^{2} is obviously Levi-flat, and since, by definition of M_{n}, its boundary is contained in M_{n}, we conclude that U_{n} × {0} ⊂ M̂_{n} ⊂ E(U × ℝ_{v}). This implies that U_{n} ⊂ E(U). The last

inclusion holds true for all n, therefore, U = ⋃_{n=1}^{+∞} U_{n} ⊂ E(U) as it was

claimed above.

To complete the proof of Theorem 5 we construct a function φ ∈ C(U) such that U is the maximal domain where the Levi-flat extension of φ can be defined. Since, in view of Theorem 4, we always have a Levi-flat extension to E(U), it will imply that E(U) ⊂ U and, hence, by the inclusion proved above, one has E(U) = U and this domain is maximal for the Levi-flat extension of φ.

For each n = 1, 2, … choose t_{n} ∈ ℝ in such a way that all discs D_{n,t_{n}}^{α} are contained in {v ≤ −n}. Further, for (z, u) ∈ bU_{n} define φ(z, u) to be equal to ImG_{n}^{α}(ζ) + t_{n}, where ζ is the point of bU_{n} such that (z, u) = (π(ζ), h^{−}(ζ)). Since the sequence of sets {bU_{n}} has no accumulation point in U, we can extend this function to a continuous function φ defined on the whole of U. We claim that U is the maximal domain for the Levi-flat extension of the defined function φ.

Assume, to get a contradiction, that there is a domain Ω⊂\overline{G}, U ⊊ Ω, such that our function φ has a continuous Levi-flat extension Φ to Ω. Consider a point Q ∈ (bU∖bG)∩Ω. Then, by definition of U, for each n = 1, 2, … we can choose α_{n} in such a way that Q will be contained in the set of accumulation points for the sequence of graphs

Γ(F_{n}^{α_{n}}) = \{(z, u) : z = π(ζ), u = F_{n}^{α_{n}}(ζ), ζ ∈ \overline{B}_{n}^{α_{n}}\}

of functions F_{n}^{α_{n}}. By our assumption, the functions Φ is continuous on the set Γ(F_{n}^{α_{n}}) ⊂ Ω. Hence, for c > 0 big enough the disc D_{n,c}^{α_{n}} will not intersect the graph Γ(Φ) of Φ. Since, by our assumption, Γ(Φ) is Levi-flat, the complement to Γ(Φ) in ℂ_{z,w}^{2} is locally pseudoconvex at the interior points of Γ(Φ). Applying “Kontinuitätssatz” to the family of discs D_{n,t}^{α_{n}}, t_{n} ≤ t ≤ c, we conclude that D_{n,t}^{α_{n}} ∩ Γ(Φ) = ∅ for all t > t_{n}. This means that the disc D_{n,t_{n}}^{α_{n}} is situated above Γ(Φ) in the v-direction. Then the definition of φ on bU_{n} and the choice of t_{n} in this definition imply that Φ ≤ −n on Γ(F_{n}^{α_{n}}). Now the fact that Q is contained in the accumulation set of the sequence {Γ(F_{n}^{α_{n}})} contradicts to our assumptions that Q ∈ Ω and Φ ∈ C(Ω). □

In view of Theorems 4 and 5 and the argument of Theorem 2 applied to the discs D_{n,t}^{α} instead of the Bedford-Klingenberg’s discs, we have the following Corollary. Under the assumptions of Theorem 5 the sets E(U_{α}) are disjoint and E(U) = ⋃_{α} E(U_{α}).

Remark. Surprisingly, the domain E(U) will not be maximal for Leviflat extensions from U if we relax our assumption that every connected component of U is contained either in bG^{−} or in bG^{+}. Namely, there is a smoothly bounded simply-connected domain U in the boundary of the unit ball 𝔹 of ℂ_{z} × ℝ_{u} such that every function φ ∈ C(U) admits a continuous Levi-flat extension to a subdomain of 𝔹 which is bigger than E(U). The authors are planning to present an example of such set U and to study this phenomena in more details in the next paper.

7. Examples.

Example 1. Let G be the solid torus in ℂ_{z} × ℝ_{u} defined by the inequality (|z| − 2)^{2} + u^{2} < 1. By a direct computation of the Leviform one can easily see that the domain G × ℝ_{v} ⊂ ℂ_{z,w}^{2} is strictly pseudoconvex (see also Theorem 1 in [CS]). Consider the open subset U = \{(z, u) ∈ bG : |z| < 2\} of bG. Then U_{c} = bG \ \overline{U} = \{(z, u) ∈ bG : |z| > 2}. The sets U and U_{c} are obviously disjoint, but the set E(U ∪ U_{c}) = {(z, u)∈G : |u| < 1} is connected (the set E(U ∪ U_{c}) × ℝ_{v} is foliated by annuli A_{C} = {(z, w) ∈ G × ℝ_{v} : u = Re C, v = Im C} where C satisfies the inequality |Re C| < 1}. □

The following open problem is the motivation to our second example.

Let 𝔻 be a strictly pseudoconvex domain in ℂ^{2} with smooth boundary. Let S be a smooth 2-dimensional sphere embedded into b𝔻. Does it follow that there is a uniquely determined 3-dimensional ball B^{3} embedded into 𝔻 and foliated by holomorphic discs such that bB^{3} = S?

Note, that by [BK] and [Kr] there are 2-spheres S′ ⊂ b𝔻 arbitrarily close to S which satisfy all the requirements of the problem. A partial answer to the raised question is given by the following Proposition 2. If Ŝ is the A(𝔻)-hull of S, then the set bŜ ∩ 𝔻 is the disjoint union of holomorphic discs with the boundary in S.

Remark. We do not know if Ŝ ∖ S is a 3-ball or, even, if the set Ŝ ∖ S has an empty interior.

Proof. Since S locally divides b𝔻 in two connected components b𝔻′ and b𝔻′′, we can consider two sequences of disjoint 2-spheres S′_{n} ⊂ b𝔻′ and S′′_{n} ⊂b𝔻′′ converging to S. It is possible to choose all the spheres S′_{n} and S′′_{n} in such a way that they satisfy the conditions of [Kr] and, hence, they bound 3-balls B′_{n} and B′′_{n}, respectively, foliated by holomorphic discs. By “Kontinuitätssatz” we can easily see that, since the spheres S′_{n} and S′′_{n} were chosen to be disjoint, all the balls B′_{n} and B′′_{n} will also be disjoint. For each n = 1, 2, … denote by M_{n} the closed connected component of 𝔻 ∖ (B′_{n} ∪ B′′_{n}) containing S on its boundary and by N_{n} the closed connected component of b𝔻 ∖ (S′_{n} ∪ S′′_{n}) containing S. Then \{M_{n}\} is a decreasing sequence of compact subsets of \overline{𝔻}, therefore

M = ⋂_{n=1}^{∞} M_{n} is also a compact subset of \overline{𝔻}. We claim that M = Ŝ

and bM ∩ 𝔻 is the disjoint union of holomorphic discs with boundary in S.

To show that Ŝ ⊂ M we observe that, by the definition of M, for each ε > 0 there are balls B′_{n} and B′′_{n} contained in the ε-neighbourhood U_{ε}(M) of M. By Lemma 1, we know that the set 𝔻 ∖ Ŝ is pseudoconvex. Hence, applying “Kontinuitätssatz” to the discs foliating B′_{n} and B′′_{n}, we conclude that Ŝ ∩ B′_{n} = ∅ and Ŝ ∩ B′′_{n} = ∅. Since, due to Shilov Idempotent Theorem (see [G], p. 88), the set Ŝ is connected, one has Ŝ ⊂ M_{n}. This argument holds true for all ε > 0, therefore

Ŝ⊂ ⋂_{n=1}^{∞} M_{n} = M.

Now we prove that bM∩𝔻 is the disjoint union of holomorphic discs with boundary in S. In paricular, by virtue of the maximum modulus principle, this will imply the other inclusion M ⊂ Ŝ and, hence, that M = Ŝ. Let Q be a point of bM ∩ 𝔻. It follows then from the definition of M that either there is a sequence of points Q_{n} ∈ B′_{n} or a sequence of points Q_{n} ∈ B′′_{n} converging to Q. Since the argument in both cases is the same, we assume that Q_{n} ∈ B′_{n}. Let D_{n} be the holomorphic disc given by [Kr] which is contained in B′_{n} and such that Q_{n} ∈ D_{n}. Since the area of D_{n} is less than the area of S′_{n} = bB′_{n}, and since the spheres S′_{n} converging to S can be chosen in such a way that the areas of S′_{n} are uniformly bounded, we know by [Bi] that there is a subsequence {D_{n_{k}}} converging to a holomorphic disc D_{Q}. It follows

then from the inclusion D_{n_{k}} ⊂M_{n_{k}} and the definition of M = ⋂_{n=1}^{∞} M_{n}

that D_{Q} ⊂ M. Since D_{n_{k}} ⊂ B′_{n_{k}}, and since, by the argument above, B′_{n_{k}} ∩ M = ∅, we conclude that D_{Q} does not intersect the interior part of M and, hence, D_{Q} ⊂ bM ∩ 𝔻. To show the unicity of the holomorphic disc contained in bM∩𝔻 and passing through Q we argue by contradiction assuming that there is another disc D ≠ D_{Q}. Since Q ∈ D ∩ D_{Q}, and since the property of two holomorphic discs in ℂ^{2} to have a nonempty intersection is stable under small perturbation, we conclude that D ∩ D_{n_{k}} ≠ ∅ for k big enough. Then the fact that M ∩ B′_{n_{k}} = ∅, D_{n_{k}} ⊂ B′_{n_{k}} and D ⊂ M gives the desired contradiction thereby proving the unicity of D_{Q}. It is easy to see that the unicity of the holomorphic disc contained in bM and passing through a given point of bM ∩ 𝔻 implies that the whole set bM ∩ 𝔻 is the disjoint union of holomorphic discs with boundary lying in S. This completes the proof of Proposition 2. □

Note, that if the domain 𝔻 is bounded and diffeomorphic to a 4ball, then S divides b𝔻 into two connected components b𝔻′ and b𝔻′′. Hence, by an argument similar to the one in part 2 of Theorem 1, we have E(b𝔻′) = E(b𝔻′) and E(b𝔻′′) = E(b𝔻′′), the sets E(b𝔻′) and E(b𝔻′′) are disjoint and Ŝ = \overline{𝔻}∖(E(b𝔻′) ∪ E(b𝔻′′)). As we have already mentioned above, we do not know if the interior part of the set Ŝ = \overline{𝔻}∖(E(b𝔻′) ∪ E(b𝔻′′)) is empty. Surprisingly, for 𝔻 being a rigid domain in ℂ^{2} this can occur. More precisely, there is an open smoothly bounded domain U, diffeomorphic to a 2-disc on the boundary b𝔹 of the unit ball 𝔹 ⊂ ℂ_{z} × ℝ_{u} such that the interior part of the set 𝔹 × ℝ_{v} ∖ (E(U × ℝ_{v}) ∪ E((b𝔹 ∖ \overline{U}) × ℝ_{v})) is nonempty. An example of such set U is presented below.

Example 2. Consider the unit ball 𝔹 in ℂ_{z} × ℝ_{u}. Then, according to our notations in Section 4, the covering model G of 𝔹 is the unit disc Δ = {|z| < 1} ⊂ ℂ_{z} and for the functions h^{±} defining b𝔹 we have h^{±}(z) = ±\sqrt{1 − |z|^{2}}.

Consider the smooth 2-disc in b𝔹 with the projections p(U^{±}) of the sets U^{±} = U ∩ b𝔹^{±} on Δ as in fig. 1.

It follows then from our definition of the sets Σ and U^{±} (see Section 4) and from the description of E(U) given by Proposition 1 that the set bE(U) ∩ 𝔹 is the union of the graphs Γ(F^{±}) of harmonic functions F^{±} defined on the corresponding domains U^{±} (see figg. 2 and 3, where the boundary data of F^{±} are also indicated) and the segment 𝔹∩{z = A_{1}}.

Similarly, the set U_{c} = b𝔹 ∖ \overline{U} is also a 2-disc with the projections p(U_{c}^{±}) of the sets U_{c}^{±} = U_{c} ∩ b𝔹^{±} on Δ as in fig. 4. The corresponding sets Σ_{c} and U_{c}^{±} are shown in figg. 5 and 6 as well as the boundary data of the functions F_{c}^{±}, whose graphs together with the segments 𝔹 ∩ {z = A_{i}}, i = 2, 3, 4, constitute the set bE(U_{c}) ∩ 𝔹. It is clear now that, since the graphs Γ(F^{±}) do not coincide with the graphs Γ(F_{c}^{±}), and since the sets E(U) and E(U_{c}) are disjoint, there is an open subdomain Ω of 𝔹, bounded by the graphs Γ(F^{±}), Γ(F_{c}^{±}) and the segments 𝔹 ∩ {z = A_{i}}, i = 1, 2, 3, 4, such that

\overline{𝔹} × ℝ_{v} ∖ (E(U × ℝ_{v}) ∪ E(U_{c} × ℝ_{v})) = Ω × ℝ_{v}.

Note also, that the boundary of the domain Ω×ℝ_{v} consists of two Leviflat surfaces: L_{1}, which is constituted by the hypersurfaces Γ(F^{±})×ℝ_{v} foliated by holomorphic discs and the strip (𝔹 ∩ {z = A_{1}}) × ℝ_{v}, and L_{2}, which is constituted by the hypersurfaces Γ(F_{c}^{±}) × ℝ_{v} foliated by holomorphic discs and the strips (𝔹 ∩ {z = A_{i}}) × ℝ_{v}, i = 2, 3, 4,. Both Levi-flat hypersurfaces L_{1} and L_{2} have the same boundary bU × ℝ_{v}. □

Finally, we formulate some questions which remains open.

Question 1. Describe the set E(U) in the intrinsic terms of the space ℂ_{z} × ℝ_{u} (i.e. without using the domain G × ℝ_{v} ⊂ ℂ_{z,w}^{2} and the hulls K̂_{n} of compacts K_{n} exhausting U × ℝ_{v}).

Question 2. For each open set U ⊂ bG find the maximal open subset E of \overline{G} such that every function φ ∈ C(U) has a continuous Levi-flat extension to E.

References

[BG] E. BEDFORD and B. GAVEAU, Envelopes of holomorphy of certain 2spheres in ℂ^{2}, Amer. J. Math. 105 (1983), 975-1009.

[BK] E. BEDFORD and W. KLINGENBERG, On the envelopes of holomorphy of a 2-sphere in ℂ^{2}, J. Amer. Math. Soc. 4 (1991), 623-646.

[Bi] E. BISHOP, Conditions for the analyticity of certain sets, Mich. Math. J. 11 (1964), 289-304.

[Br] H. J. BREMERMANN, Die Holomorphiehüllen der Tuben- und Halbtubengebiete, Math. Ann. 127 (1954), 406-423.

[CS] E. M. CHIRKA and N. V. SHCHERBINA, Pseudoconvexity of rigid domains and foliations of hulls of graphs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), XXI (1995), 707-735.

[El] Y. ELIASHBERG, Filling by holomorphic discs and its applications, London Math. Soc. Lecture Note Ser., 151 (1991), 45-67.

[Er] O. G. EROSHKIN, On a topological property of the boundary of an analytic subset of a strictly pseudoconvex domain in ℂ^{2}, Mat. Zametki 49 (1991), 149-151.

[G] T. W. GAMELIN, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969.

[Ke] N. KERZMAN, Hölder and L^{p} estimates for solutions of ¯∂u = f in strongly pseudoconvex domains, Comm. Pure Appl. Math. 24 (1971), 301-379.

[Ki] C. KISELMAN, The partial Legendre transformation for plurisubharmonic functions, Invent. Math. 49 (1978), 137-148.

[Kr] N. G. KRUZ̆ILIN, Two-dimensional spheres in the boundary of strictly pseudoconvex domains in ℂ^{2}, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 1194-1237.

[N] R. NARASIMHAN, The Levi problem for complex spaces. II, Math. Ann. 146 (1962), 195-216.

[Sh] N. SHCHERBINA, On the polynomial hull of a graph, Indiana Univ. Math. J. 42 (1993), 477-503.

[ST] N. SHCHERBINA and G. TOMASSINI, The Dirichlet problem for Leviflat graphs over unbounded domains, Internat. Math. Res. Notices (1999), 111-151.

[St] E. L. STOUT, Removable singularities for the boundary values of holomorphic functions, Several complex varuables (Stockholm, 1987/1988), 600-629, Math. Notes, 38, Princeton Univ. Press, Princeton, NJ, 1993.

[TT] E. CASADIO TARABUSI and S. TRAPANI, Envelopes of holomorphy of Hartogs and circular domains, Pacific J. Math. 149 (1991), 231-249.

[T] G. TOMASSINI, Sur les algèbres A^{0}(\overline{D}) et A^{∞}(\overline{D}) d’un domaine pseudoconvexe borné, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), X (1983), 243-256.

Nikolay Shcherbina Department of Mathematics University of Göteborg 412 96 Göteborg

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