On some class of sets with multi-sheeted envelope of holomorphy

B. Jöricke, N. Shcherbina

Abstract. Let m be a smooth CR-manifold of dimension 2n − 2 contained in the unit sphere ∂𝔹^{n} in ℂ^{n}, n > 2. We prove that if ∂M is the finite union of boundaries of (n − 1)-dimensional analytic varieties and M is globally minimal, then the envelope of holomorphy of ∂𝔹^{n} \ m̄ is multi-sheeted. In some cases we prove that the number of sheets is exactly two. It is also proved (for 4-dimensional CR-manifolds contained in ∂𝔹^{3} ) that the property of being globally minimal (even being of finite type at each point) is generic.

For general domains in ℂ^{n} it is difficult to understand from the geometry of the domain whether its envelope of holomorphy is single-sheeted over ℂ^{n}. It has become a fruitful attempt to consider the mentioned problem for suitably small neighbourhoods (or one-sided neighbourhoods) of subsets of the unit sphere in ℂ^{n} (more general, of subsets of strictly pseudoconvex boundaries in ℂ^{n}).

Bedford and Klingenberg [BK] (and later Kružilin [K]) proved that the envelope of holomorphy of generic two-spheres in strictly pseudoconvex boundaries in ℂ^{2} (in the sense of the space of maximal ideals of the algebra of germs of analytic functions near the two-sphere) is a three-ball which is foliated by analytic discs, in particular, the envelope of holomorphy is single-sheeted.

The case of open subsets of strictly pseudoconvex boundaries in ℂ^{2} was first considered by Stout [St1]. Let G ⊂ ℂ^{n} be a strictly pseudoconvex domain and let ω ⊂ ∂G be an open subset of the boundary with connected components ω_{1}, ω_{2}, . . .. A connected open subset U_{j} is called a one-sided neighbourhood of ω_{j} if U_{j} is the intersection of G with an open subset of ℂ^{n} which intersects ∂G along ω_{j}. If U_{1}, U_{2}, … are disjoint one-sided neighbourhoods of ω_{1}, ω_{2}, … then ∪_{j}U_{j} is called a one-sided neighbourhood of ω. Using local analytic extension from (neighbourhoods of) hypersurfaces near strictly pseudoconvex points, one can choose a one-sided neighbourhood U of ω such that any analytic function in an arbitrary one-sided neighbourhood of ω (and also any continuous CR-function on ω) has analytic extension to U. Call such sets U one-sided neighbourhoods of ω with a priori analytic continuation property. The envelope of holomorphy of such a set U (more precisely, the disjoint union of envelopes of holomorphy of its connected components) is called the essential envelope of holomorphy of ω and is denoted by E(ω). (It is easy to see that E(ω) does not depend on the choice of the set U with the above mentioned property.)

Lupacciolu-Stout, Alexander-Stout and Slodkowski (see e.g. the survey paper [St2]) proved that for any bounded strictly pseudoconvex domain G ⊂ ℂ^{2} and any open subset ω of the boundary ∂G the essential envelope of holomorphy E(ω) is single-sheeted, moreover, it coincides with G \ K̂_{A(G)}, where K = ∂G \ ω and K̂_{A(G)} = {z ∈ Ḡ : |f(z)| ≤ max_{K} |f| for each f ∈ A(G)} is the hull of K with respect to the algebra A(G) of holomorphic functions in G which are continuous in the closure Ḡ.

For dimensions larger than 2 the situation is different. Čirka and Stout [ČSt] were the first who constructed a strictly pseudoconvex domain G in ℂ^{4} (the union of the unit ball with a tubular neighbourhood of a simple smooth arc which joins two distinct points in the unit sphere) and an open subset ω of ∂G with twosheeted essential envelope of holomorphy E(ω).

In this paper we consider the case of “large” open subsets ω of strictly pseudoconvex boundaries, more precisely, of complements of subsets of codimension one submanifolds of such boundaries. Let G ⊂ ℂ^{n}, n > 2, be an arbitrary bounded strictly pseudoconvex domain with C^{2} boundary and let M ⊂ ∂G be a connected generic CR-manifold of class C^{2} and of dimension 2n−2 (hence of CR-dimension n − 2). A subset A of M is called CR-invariant if it is the union of some CRorbits of M. It is proved in [J2] that a compact subset K of M is removable (i.e. E(∂G \ K) = G) if K does not contain non-empty CR-invariant subsets of M. Moreover, for any compact K ⊂ M the equality E(∂G \ K) = E(∂G \ K̃) holds, where K̃ is the largest compact CR-invariant subset of M contained in K. The following problem was raised in [J2].

Problem. For which compact CR-invariant subsets K of M the essential envelope of holomorphy E(∂G \ K) is multi-sheeted?

It is natural to expect that this happens always if K contains a CR-orbit of M of full dimension. Our theorem 1 gives a partial answer to this problem.

Theorem 1. Let M be an orientable generic CR-manifold of real dimension 2n − 2 and of class C^{2} contained in the unit sphere ∂𝔹^{n} of ℂ^{n}, n > 2. Let K ⊂ M be a compact CR-invariant subset of M such that

1. ∂K = ∪_{j=1}^{ℓ}N_{j} is the disjoint union of finitely many connected compact maximally complex CR-manifolds N_{j} of dimension 2n − 3 and of class C^{2}.

2. K contains a CR-orbit m of M of dimension 2n − 2.

Then the envelope of holomorphy E(∂𝔹^{n} \ K) is multi-sheeted. Moreover, each point of m is a limit of points over which E(∂𝔹^{n} \ K) is multi-sheeted.

Recall that a CR-manifold is called globally minimal if any pair of its points can be joined by a piecewise CR-curve.

Remark 1. Condition 2 of the theorem is satisfied in particular if the following condition 2′ holds.

2′. The interior of K (with respect to M), intK, is globally minimal and, hence, it is itself a CR-orbit of M.

If the conditions of theorem 1 are satisfied with condition 2 replaced by the stronger condition 2′, then the whole set K is contained in the closure of the set of points over which E(∂𝔹^{n} \ K) is multi-sheeted.

Remark 2. The proof of theorem 1 applies without modification for 𝔹^{n} replaced by any strictly pseudoconvex domain G ⊂ ℂ^{n}, n > 2, with C^{2} boundary such that H^{2}(G, ℤ) = 0.

Remark 3. Under the condition of theorem 1 the N_{j} are compact CR-orbits of M of real codimension 1. Each N_{j} is orientable, hence by the theorem of Harvey and Lawson [HaL], N_{j} bounds an analytic variety X_{j} ⊂ 𝔹^{n} of complex dimension n − 1. We will use the condition that the N_{j} are of class C^{2}. If we require that M is e.g. of class C^{2,α} this is satisfied automatically.

The relation of the theorem to the posed problem is the following. Suppose M is an orientable generic CR-manifold of real dimension 2n − 2 and of class C^{2,α} contained in a strictly pseudoconvex boundary in ℂ^{n}, n > 2. Let K be a CR-invariant compact subset of M which contains a CR-orbit m of M of full dimension. Then the boundary ∂m of m is the union of CR-orbits of codimension 1 in M, hence each such orbit either bounds an analytic variety of dimension n−1 or it is an exceptional orbit (see [J2] for more details). So the full answer to the problem is, in particular, related to the question whether non-empty compact exceptional minimal CR-invariant subsets of generic (2n − 2)-dimensional CRmanifolds contained in strictly pseudoconvex boundaries in ℂ^{n}, n > 2, are always non-removable.

The natural question arises whether it is a generic property of CR-manifolds to be globally minimal. The following proposition gives the positive answer to this question in the situation which is interesting for us here. It states, in particular, that CR-invariant compact subsets of sufficiently smooth 4-dimensional CR-manifolds in strictly pseudoconvex boundaries in ℂ^{3} which enjoy property 1 of theorem 1 generically also have property 2′, moreover, generically for such a CR-manifold each point is of finite type and, hence, minimal in the sense of Tumanov. For simplicity the assertion is stated and proved only for dimension n = 3. We believe that the same methods allow to prove the corresponding result for arbitrary dimensions. We also believe that similar arguments but more extended calculation (and a clever induction) allows to prove, for example, that each generic CR-manifold of class C^{∞} of arbitrary codimension in ℂ^{n}, (n ≥ 2 arbitrary), has arbitrarily small deformations which are globally minimal, moreover, each point of which is of finite type.

Proposition. Let G be a strictly pseudoconvex domain in ℂ^{3} with C^{∞}-smooth boundary ∂G. Let M be an orientable C^{4}-smooth generic CR-submanifold of bG of real dimension 4 and of CR-dimension 1. Let K be a compact CR-invariant subset of M. Then there are arbitrarily small C^{4}-deformations M̃ of M in ∂G which fix the set M \ intK (here intK is the set of all interior points of K with respect to M) and such that the deformation intK̃ of the set intK is C^{∞}-smooth and globally minimal. Moreover, it is of type 4, more precisely, the Lie brackets of order up to 3 of two real vector fields generating T_{p}^{ℂ}(intK̃) span the whole tangent space T_{p}(intK̃) for all p ∈ intK̃.

The following two examples illustrate theorem 1 and proposition.

Example 1. Let M = \{z ∈ ∂𝔹^{3} : z_{1} ∈ (−\frac{1}{2}, \frac{1}{2})\}. It is not hard to check that M is a generic CR-manifold of real dimension 4 (here n = 3, 2n − 2 = 4). The boundary ∂M = \{−\frac{1}{2}\} × \{|z_{2}|^{2} + |z_{3}|^{2} = 1 − \frac{1}{4}\} ∪ \{\frac{1}{2}\} × \{|z_{2}|^{2} + |z_{3}|^{2} = 1 − \frac{1}{4}\} satisfies condition 1 of the theorem. Hence there exist arbitrarily small (in C^{4} topology) C^{∞} deformations M̃ of M with fixed ∂M̃ = ∂M such that ∂𝔹^{3}\ M^{≂} has multi-sheeted essential envelope of holomorphy E(∂𝔹^{3}\ M^{≂}).

Example 2. Let c < \frac{1}{\sqrt{3}^{3}} and close to \frac{1}{\sqrt{3}^{3}} . The set N = \{z = (z_{1}, z_{2}, z_{3}) ∈ ∂𝔹^{3} : z_{1}z_{2}z_{3} = c} is a smooth compact connected 3-dimensional maximally complex CR-manifold. Using Gromov’s h-principle one can see that N bounds a smooth generic 4-dimensional CR-manifold M^{∗} contained in ∂𝔹^{3} (see also [JSh] where this fact is proved in an elementary way). By theorem 1 and proposition, for a smooth deformation M̃ of M^{∗} with ∂M̃ = ∂M^{∗} the envelope of holomorphy E(∂𝔹^{3}\ M^{≂}) is multi-sheeted. Note that in this case M̃ has connected boundary.

In fact, in both cases the essential envelope of holomorphy is exactly twosheeted provided the manifold M̃ is close enough to a manifold which admits a foliation (maybe, with singularities) into maximally complex CR-manifolds of dimension 2n − 3. We postpone the proof of this fact until the end of the paper.

Proof of theorem 1. 1. Here we recall a result from the theory of CRmanifolds which we will use in the proof. Let m ⊂ ℂ^{n} be the manifold which appeared in condition 2 of theorem 1. Recall that a wedge W with edge m is a set of the form W = {x + y ∈ ℂ^{n} : x ∈ m, y ∈ K(x)} where K(x) is an open truncated cone in ℂ^{n} with vertex x. The manifold m is a generic globally minimal CR-manifold of class C^{2} and codimension one in a strictly pseudoconvex boundary in ℂ^{n}, hence there exists a wedge W with edge m such that any continuous CR-function on m has analytic continuation to W. Moreover, the following stability assertion holds.

Let m′ be a manifold which is close enough to m in C^{2} (i.e. m′ is obtained by moving m along a C^{2} small vector field defined near m). Then clearly m′ is again a generic CR-manifold. Moreover, the same analytic continuation property takes place for a wedge W′ with edge m′ such that W′ is close to W (in the sense that the parameters of corresponding truncated cones K(x) and K(x′) at related points x ∈ m, x′ ∈ m′, are uniformly close for vertices x ranging over compact subsets of m).

For the wedge extendibility result for globally minimal generic CR-manifolds of class C^{2,α} and of arbitrary codimension in ℂ^{n} see [M], [J1]. In the case of class C^{2} see [J3], where also the stability result is shown (see the proof of proposition 4.1 there). For the smoothness class C^{2,α} the stability result follows directly from known proofs of wedge-extendibility. Indeed, the main tool for proving wedgeextendibility is to attach analytic discs consecutively to the union of the CRmanifold with suitable sets which are covered by already constructed analytic discs. The analytic discs are contained in the set to which functions extend analytically and cover a wedge at final step. The construction of the mentioned analytic discs uses Bishop’s equation for suitable parameters. These equations are solvable if the CR-manifold is of class C^{2,α}. Moreover, they remain to be solvable after small perturbation and the solutions depend continuously on the perburbation. This implies the stability result.

2. Here we fix notation and introduce coordinates in a spherical layer around ∂𝔹^{n} which will be convenient for the proof.

Each connected component N_{j} of ∂K bounds an (n − 1)-dimensional analytic variety V_{j}, moreover, in a neighbourhood of ∂𝔹^{n} each V_{j} is a C^{2} manifold with C^{2}-boundary N_{j}. (It follows by locally presenting V_{j} as graph of a CR-function of class C^{2} over a piece of a strictly pseudoconvex hypersurface in ℂ^{n−1} and using analytic extension of the graphing function). By Forstnerič [F], T_{p}N_{j} is not contained in the complex tangent space T_{p}^{c}∂𝔹^{n} for each p ∈ N_{j}, i.e., V̄_{j} is transversal to ∂𝔹^{n}. Hence there exists a non-singular vector field v defined in a neighbourhood of the union ∪_{j=1}^{ℓ}N_{j} which is for each j tangental to V̄_{j} at points of V̄_{j} and is transversal to r∂𝔹^{n} for all r close to 1. Normalizing, we may assume that at each point p the equality v(p) = n(p) + τ(p) holds, where n(p) is the unit inner normal to |p|∂𝔹^{n} = ∂(|p|𝔹^{n}) (the sphere around zero to which p belongs) and τ(p) is orthogonal to n(p). Extend v to a vector field with the latter properties in a spherical layer {z ∈ ℂ^{n} : 1 − ε < |z| < 1 + ε} for some ε > 0 and denote the extended vector field again by v. For each t ∈ [0, ε) the flow Φ_{t} of v maps ∂𝔹^{n} diffeomorphically onto (1 − t)∂𝔹^{n}, moreover, it maps each N_{j} ⊂ ∂𝔹^{n} diffeomorphically onto V_{j} ∩(1−t)∂𝔹^{n} and maps ∂𝔹^{n} \ ∪_{j=1}^{ℓ}N_{j} diffeomorphically onto (1 − t)∂𝔹^{n} \ ∪_{j=1}^{ℓ}V_{j}. For t ∈ (0, ε) denote

M_{1−t} =^{def} Φ_{t}(M), K_{1−t} =^{def} Φ_{t}(K), m_{1−t} =^{def} Φ_{t}(m). In particular, K_{1−t} ⊂ (1−t)∂𝔹^{n}, it is diffeomorphic to K and its boundary equals (1 − t)∂𝔹^{n} ∩ ∪_{j=1}^{ℓ}V_{j}. Define E = ∪_{1−ε<r<1}M_{r}, e = ∪_{1−ε<r<1}intK_{r}, ē = ∪_{1−ε<r<1}K_{r}. (Here intK denotes the interior of K with respect to M.) Then E and e are orientable C^{2} hypersurfaces in 𝔹^{n} \ (1 − ε)\overline{𝔹^{n}} ⊂ ℂ^{n}, ē is a hypersurface with boundary which is closed in 𝔹^{n} \ (1 − ε)\overline{𝔹^{n}}. Denote U′ = (𝔹^{n} \ (1 − ε)\overline{𝔹^{n}}) \ ē. U′ is open and defines a onesided neighbourhood of ∂𝔹^{n} \ K (which is diffeomorphic to (∂𝔹^{n} \ K) × (0, ε)). It will be convenient to use the flow Φ_{t} and the set U′ in the sequel.

3. Let U be a one-sided neighbourhood of ∂𝔹\K such that analytic functions in any one-sided neighbourhood of ∂𝔹^{n} \ K extend to U. We may assume that U ⊂ U′, in particular, U does not intersect ∪_{j=1}^{ℓ}N_{j}. The following lemma holds.

Lemma 1. There exist manifolds m^{+} and m^{−} contained in U which are arbitrarily close in C^{2} to m with the same boundary ∂m^{+} = ∂m^{−} = ∂m and are situated on different sides of E in 𝔹^{n} \ (1 − ε)\overline{𝔹^{n}}. There are wedges W^{+} and W^{−} with edges m^{+} and m^{−}, respectively, contained in 𝔹^{n} which are close to W if m^{+} and m^{−} are close to m such that the following holds. Each analytic function in U has analytic extension from m^{+} to W^{+} and from m^{−} to W^{−}. In other words, W^{−} and W^{+} embed naturally into the essential envelope of holomorphy E(∂𝔹^{n} \ K).

Lemma 1 together with the following lemma imply the theorem.

Lemma 2. There exists an analytic function F in U′ ⊃ U such that the following holds. There exist intersecting wedges W^{+} and W^{−} as in lemma 1 with edges m^{+} and m^{−} contained in U such that the analytic extensions of F to W^{+} and to W^{−}, respectively, differ on a non-empty open subset of the intersection W^{+}∩W^{−}. More precisely, for each point q ∈ m the wedges may be chosen so that the respective analytic continuations of F to intersecting open truncated cones K^{+}(q(m^{+})) ⊂ W^{+} and K^{−}(q(m^{−})) ⊂ W^{−} differ a.e. on their intersection. Moreover, the points q(m^{+}) ∈ m^{+} and q(m^{−}) ∈ m^{−} can be taken close to q for W^{+} and W^{−} close to W.

Proof of Lemma 1. Since M is orientable, there is a non-singular C^{2} vector field ν in a neighbourhood of M which is transversal to M and tangential to ∂𝔹^{n}. Let φ be a small non-negative C^{2} function in a neighbourhood of M which vanishes exactly on M \ m. Consider the flow of the vector fields φ · ν and −φ · ν and apply the time 1 mapping to m (and to m̄). We obtain two CR-manifolds contained in ∂𝔹^{n} \ m̄, which are C^{2} close to m, have the same boundary as m and are situated locally on different sides of m in ∂𝔹^{n}. Move each of the two manifods slightly in the direction of the vector field v defined in part 2 of the proof, precisely, apply to each of the manifolds the time 1 mapping of the flow ψv for a small non-negative C^{2} function ψ in a neighbourhood of M, which vanishes on M \ m and is chosen in such a way that we obtain two manifolds m^{−} and m^{+} contained in U. If φ and ψ are small, then m^{−} and m^{+} are close to m. Moreover, ∂m^{−} = ∂m^{+} = ∂m. The existence of the required wedges W^{+}, W^{−} follows from part 1 of the proof. By pseudoconvexity of 𝔹^{n} both are contained in 𝔹^{n}. □

Proof of Lemma 2. Observe first that for each j there exists an analytic function f_{j} in 𝔹^{n} with simple zero on V_{j} and no other zeros. (f_{j} is a minimal defining function in the terminology of [Č1].) It is characterized by the property that (locally) it divides any other analytic function which vanishes (locally) on V_{j}. An equivalent characterization is that the gradient of the function does not vanish at regular points of the variety (see [Č1]). Locally such a function exists since V_{j} is an irreducible analytic variety of pure dimension n − 1. The global existence of the f_{j} follows from solvability of the second Cousin problem in 𝔹^{n}.

Put f = ∏_{j=1}^{ℓ} f_{j}. Let p ∈ U′ and let F_{p} be a holomorphic branch of \sqrt{f} defined on a small ball around p (in other words, F_{p} is a functional element at p which defines a branch of \sqrt{f}). It is clear that F_{p} has unrestricted analytic continuation in 𝔹^{n} \ ∪_{j=1}^{ℓ}V_{j} (i.e., it has analytic continuation along any curve there). We will show that F_{p} extends to an analytic function F in U′ which satisfies the requirements of lemma 2.

Prove first single-valued analytic continuation of F_{p} to U′. Put r = |p| ∈ (1 − ε, 1). By the product structure of U′ it is enough to prove that F_{p} extends to an analytic function F^{r} in a neighbourhood of r∂𝔹^{n} \ K_{r}. Indeed, suppose this has been done. Consider an arbitrary point z ∈ U′. Then z = Φ_{1−|z|}(ζ) for some ζ ∈ ∂𝔹^{n} \ K. Consider the flow line Φ_{t}(ζ), t ∈ (0, ε). It intersects |z|∂𝔹^{n} at z and r∂𝔹^{n} at some point η. Provide analytic continuation of F^{r} from a small ball around η along the flow line t → Φ_{t}(ζ). (The flow line does not meet ∪_{j=1}^{ℓ}V_{j}, so this analytic continuation is possible.) We obtain an analytic function in a small ball around z (a functional element at z). By the properties of the flow Φ_{t}, the functional elements at points of U′ obtained in the above described way constitute an analytic function in U′ which coincides with F^{r} near r∂𝔹^{n} \ K_{r}.

By a slight abuse of languge in the following we will call two functional elements at a point p equal, if they are equivalent, i.e. if they define the same function in a small enough ball around p. The existence of the analytic function F^{r} follows now directly from the following lemma.

Lemma 3. Take an arbitrary point z ∈ r∂𝔹^{n}\K_{r} and an arbitrary smooth simple closed curve γ ⊂ r∂𝔹^{n} \ K_{r} which starts at z. Let F_{z} be a functional element at z which defines some branch of \sqrt{f} in some ball around z. Then analytic continuation of F_{z} along γ leads to the same functional element at z.

Proof. The curve γ bounds a smoothly embedded disc Δ ⊂ r∂𝔹^{n}. Indeed, removing from r∂𝔹^{n} a point which is not contained in γ and choosing coordinates on the punctured sphere we may assume that γ is obtained by a smooth embedding of the unit circle 𝕋 = ∂𝔻 ⊂ ℂ into ℝ^{2n−1}. Extend this mapping to a smooth mapping of the closed disc 𝔻̄ into ℝ^{2n−1} whose restriction to a small annulus attached to the circle 𝕋 is an embedding. By Wintney’s embedding theorem (see e.g. [Hi] Ch. 2) this mapping can be approximated by an embedding of the disc 𝔻. (Here we use that 2n − 1 exceeds twice the real dimension of 𝔻, since n > 2.) Moreover, (by a partition of the unit on 𝔻) we may assume that this embedding coincides with the previous mapping near 𝕋. This proves the existence of Δ. Endow Δ with the orientation induced from 𝔻 by the embedding.

Let M̃_{r} be a small neighbourhood of K_{r} on M_{r} which still does not meet γ = ∂Δ. (Recall γ ⊂ r∂𝔹^{n} \ K_{r}.) By the transversality theorem [Hi, Th. 3.2.1], there is a small smooth perturbation of Δ inside r∂𝔹^{n} which fixes ∂Δ = γ and is transversal to M̃_{r} and to each of the V_{j} ∩ r∂𝔹^{n}. Denote this perturbation again by Δ. Then Δ ∩ ∪_{j=1}^{ℓ}V_{j} consists of isolated points and Δ ∩ M̃_{r} consists of curves. Hence the compact set Δ ∩ K_{r} is the finite union of disjoint compact (connected) curves, Δ ∩ K_{r} = ∪_{k}s_{k} ∪ ∪_{k′}c_{k′}, where each c_{k′} is a loop, i.e. a simple closed C^{2} curve, and s_{k} is a closed segment, i.e. a simple C^{2} curve with two boundary points belonging to ∪_{j=1}^{ℓ}V_{j}. Note that Δ intersects the union of the V_{j} only at the endpoints of the segments s_{k}. In particular, the loops c_{k′} do not meet ∪_{j=1}^{ℓ}V_{j}. Note that F_{z} has analytic continuation along any curve in Δ̄ \ ∪_{j=1}^{ℓ}V_{j}, in particular, by the remark just made, F_{z} has analytic continuation along any curve in Δ̄ \ ∪_{k}s_{k}.

To prove the lemma, it is enough to prove that F_{z} extends to a single-valued analytic function in a neighbourhood of Δ̄ \ ∪_{k}s_{k}. For this it is enough, as in the classical situation of the monodromy theorem, to prove the following assertion. For each k let γ_{k} be a simple closed Jordan curve in Δ which surrounds exactly one of segments s_{k} (i.e. the corresponding s_{k} are contained in the connected component of Δ̄\γ_{k} which does not contain ∂Δ, the other segments are contained in the other component). Let z_{k} be a point of γ_{k}. Then lemma 3 holds for γ and z replaced by γ_{k} and z_{k}.

Fix now k and prove the assertion. Since any branch of \sqrt{f} extends to interior points of s_{k} from both sides in Δ, we may consider instead of a closed Jordan curve surrounding s_{k} the following curve γ_{k} which is the limit of such Jordan curves.

Denote the endpoints of s_{k} by a_{k} and b_{k}. Then a_{k} ∈ V_{j(a_{k})}, b_{k} ∈ V_{j(b_{k})} for some indices j(a_{k}) and j(b_{k}). Consider a small circle in Δ around a_{k} and denote its intersection with s_{k} by â_{k}. Analogously, consider a small circle in Δ around b_{k} and let b̂_{k} be its intersection point with s_{k}. Let ŝ_{k} be the part of s_{k} between â_{k} and b̂_{k}. Denote by ŝ_{k}^{+} (the plus edge of ŝ_{k}) an identical copy of the curve ŝ_{k} which is considered as limit of curves situated “on the right of s_{k}”. On ŝ_{k}^{+} we consider analytic continuation of the related branches of \sqrt{f} “from the right of ŝ_{k}”. Respectively, define ŝ_{k}^{−} (the minus edge of ŝ_{k}) to be a copy of the curve −ŝ_{k} (the curve ŝ_{k} with inversed orientation) with analytic continuation of branches of \sqrt{f} “from the left of ŝ_{k}” (i.e. from the right of −ŝ_{k}). To each point ζ in ŝ_{k} corresponds a point ζ^{+} in the plus edge ŝ_{k}^{+} and a point ζ^{−} in the minus edge ŝ_{k}^{−}. Consider the point z_{k} = â_{k}^{−} (the point on the minus edge ŝ_{k}^{−} which corresponds to â_{k}). Let α_{k} surround a_{k} along the above-mentioned circle in Δ counterclockwise (with respect to orientaiton introduced from 𝔻) starting from the point â_{k}^{−} and terminating at the point â_{k}^{+}. Let β_{k} surround b_{k} along the other circle in Δ counterclockwise starting at b̂_{k}^{+} and terminating at b̂_{k}^{−}. Let γ_{k} be the sum of the curves α_{k}, ŝ_{k}^{+}, β_{k}, ŝ_{k}^{−} in the just mentioned order.

Consider a functional element at â_{k} which defines a branch of \sqrt{f} and denote it by F_{}_{â_{k}^{−}}. We will study now analytic continuation of F_{}_{â_{k}^{−}} along γ_{k}. Recall that a_{k} ∈ V_{j(a_{k})} and is close to ∂𝔹^{n}, so it is a regular point of V_{j(a_{k})}. Hence there is a neighbourhood U(a_{k}) of a_{k} in which f has the form f = f_{j(a_{k})} · g_{j(a_{k})}, where g_{j(a_{k})} does not have roots in U(a_{k}) and V_{j(a_{k})} ∩ U(a_{k}) is the zero set of f_{j(a_{k})} in U(a_{k}). Moreover, df_{j(a_{k})} does not vanish on this set and hence (since f_{j(a_{k})} is analytic) has real rank 2 at each point of this set.

Moreover, V_{j(a_{k})} and Δ are transversal, i.e. the kernel of df_{j(a_{k})} computed at the point a_{k} and the tangent space T_{a_{k}}Δ are transversal (as real linear subspaces of T_{a_{k}}(r∂𝔹^{n})). Hence, df_{j(a_{k})} computed at the point a_{k} and restricted to T_{a_{k}}Δ is a linear isomorphism. This implies that, provided U(a_{k}) is small enough, the restriction of f_{j(a_{k})} on Δ ∩ U(a_{k}) is a diffeomorphism. Moreover, f_{j(a_{k})}(a_{k}) = 0. We may assume that the curve α_{k} is contained in Δ ∩ U(a_{k}) together with the small disc on Δ bounded by α_{k}. f_{j(a_{k})} maps the small closed disc diffeomorphically onto an open set in ℂ which contains 0 and is bounded by a simple closed Jordan curve (of class C^{2}). In particular, the curve f_{j(a_{k})} ∘ α_{k} surrounds 0 once (either counterclockwise or clockwise). This means that f_{j(a_{k})} changes argument by 2π or by −2π along α_{k} (running from â_{k}^{−} to â_{k}^{+} counterclockwise). Hence \sqrt{f} = \sqrt{f_{j(a_{k})}} · \sqrt{g_{j(a_{k})}} changes argument by +π or −π along α_{k}. We obtained two functional elements F_{}_{â_{k}^{−}} and F_{}_{â_{k}^{+}} at â_{k} such that F_{}_{â_{k}^{+}} is the analytic continuation of F_{}_{â_{k}^{−}} along α_{k}, and the two functional elements differ by the factor e^{±iπ} = −1.

Provide analytic continuation of F_{}_{â_{k}^{−}} and F_{}_{â_{k}^{+}} along ŝ_{k}^{−} in backward direction and along ŝ_{k}^{+}, respectively. For any point ζ ∈ ŝ_{k} we obtain functional elements F_{ζ^{+}} and F_{ζ^{−}} on the plus edge and on the minus edge, respectively, which differ by the factor −1 (i.e. by sign). Repeat the arguments done for the point a_{k} and the curve α_{k}, with respect to the point b_{k} and the curve β_{k}. We assume that β_{k} is contained in a sufficiently small neighbourhood of b_{k}, and obtain that analytic continuation of Fb̂_{k}^{+} along β_{k} leads to a functional element at b̂_{k}^{−} which differ from Fb̂_{k}^{+} by sign and, hence, equals Fb̂_{k}^{−}.

Summarizing, we proved that analytic continuation of F_{}_{â_{k}^{−}} along γ_{k} leads to the same functional element, which proves lemma 3. □

For the proof of lemma 2 we have to show that analytic continuations of F to some wedges W^{+} and W^{−}, found according to lemma 1, differ on some part of W^{+} ∩ W^{−}.

First, associate with the hypersurface e = ∪_{1−ε<r<1}intK_{r} its plus copy e^{+} and its minus copy e^{−} (so that the edge m^{+} of W^{+} is on the side of e^{+}). We claim that for each ζ ∈ e analytic continuation of F from U′ to the corresponding points ζ^{+} ∈ e^{+} and ζ^{−} ∈ e^{−} give functional elements F_{ζ^{+}} and F_{ζ^{−}} at ζ which differ by sign. Indeed, let z ∈ r∂𝔹^{n} ∩ V_{j} for some r ∈ (1 − ε, 1) and some j. As above, we choose a small smoooth disc Δ which contains z, is contained in r∂𝔹^{n} and is transversal to V_{j} and to M_{r}. Δ intersects K_{r} along a small curve with endpoint z in Δ. Let α be a curve in Δ which surrounds z once, starts at some point ζ_{0}^{−} ∈ m_{r}^{−} ∩ Δ and terminates at ζ_{0}^{+} ∈ m_{r}^{+} ∩ Δ. Then F_{ζ_{0}^{+}} is obtained from F_{ζ_{0}^{−}} by analytic continuation along α and hence, as above, F_{ζ_{0}^{+}} and F_{ζ_{0}^{−}} differ by sign. Analytic continuation of F_{ζ_{0}^{+}} and F_{ζ_{0}^{−}}, respectively, along curves in e gives the required claim.

Choose now manifolds m^{+} and m^{−} contained in U on the plus side and the minus side of e, respectively, and close in C^{2} to m. (Recall m is a globally minimal CR-orbit of M of dimension 2n − 2 contained in K.) Let q ∈ m and let K(q) be a truncated cone with vertex q contained in the wedge W. Note that K(q) ⊂ 𝔹^{n}. Shrinking it we may assume that it is at positive distance from ∪_{j=1}^{ℓ}V̄_{j} and contained in 𝔹^{n} \ (1 − ε)\overline{𝔹^{n}}. We may also assume (choosing m^{+} and m^{−} close enough to m) that there are points q(m^{+}) ∈ m^{+} and q(m^{−}) ∈ m^{−} close to q and truncated cones K(q(m^{+})) ⊂ W^{+} and K(q(m^{−})) ⊂ W^{−} so close to K(q) that the following holds: The two cones intersect, are contained in 𝔹^{n} \ (1 − ε)\overline{𝔹^{n}} and do not meet ∪_{j=1}^{ℓ}V̄_{j}. Join q(m^{−}) and q(m^{+}) by a curve σ which consists of two straight line pieces, one of them contained in {q(m^{−})} ∪ K(q(m^{−})), the other one in {q(m^{+})} ∪ K(q(m^{+})). We assume that the curve intersects e transversally and exactly once. Denote the intersection point by p. Then σ \ {p} is contained in U′ where F is analytic. Let σ^{+} be the component of σ \ {p} which contains q(m^{+}) and σ^{−} the other component. Then σ^{+} is in U′ on the plus side of e. Hence analytic continuation of F from m^{+} along −σ^{+} gives the functional element F_{p^{+}}. Respectively, analytic continuation of F from m^{−} along σ^{−} gives the functional element F_{p^{−}} which differs from F_{p^{+}} by sign. Hence, analytic continuation of F from m^{+} to K(q(m^{+})) ⊂ W^{+} cannot coincide with analytic continuation of F from m^{−} to K(q(m^{−})) ⊂ W^{−}. Lemma 2 (and hence theorem 1) is proved. □

Proof of proposition. First, we consider an arbitrary small C^{4}-deformation M^{≈} of M in ∂G which fixes the set M \intK such that the deformation K of the set intK is C^{∞}-smooth. Since the manifold M was assumed to be generic, and since the property of a manifold of being generic is stable under small deformations, we can assume that the manifold M^{≈} is also generic.

Consider the normal bundle N of K in ∂G. Denote by F the natural diffeomorphism of a small tubular neighbourhood of K in N onto an open subset of ∂G. The mapping F gives a correspondence of smooth sections of N close to the zero-section and smooth 4-manifolds which are small deformations of K in ∂G. Let J^{4}(K, N) be the set of all 4-jets of germs of sections of N. Denote by π the natural projection of J^{4}(K, N) onto K. Let p be a point of K and let s be a germ of a smooth section of N defined in a neighbourhood of p which is close enough to the zero-section. Then K_{s} =^{def} F (s) is a germ of a smooth hypersurface in ∂G. Let Z be a (1,0)-vector field which is complex tangential to K_{s}. Then the real vector fields ReZ, ImZ, Im[Z, Z̄], Im[Z, [Z, Z̄]], Im[Z, [Z, [Z, Z̄]]] and Im[Z̄, [Z, [Z, Z̄]]] are tangential to K_{s}. Moreover, since G is strictly pseudoconvex, it follows that the first three vector fields are linearly independent (see also (7) below). Define a subset N of J^{4}(K, N) as follows: the fiber N_{p} = N ∩ π^{−1}(p) of the set N over the point p is defined to be the equivalence class of 4-jets of germs of sections s as described above, for which the vectors Im[Z, [Z, Z̄]], Im[Z, [Z, [Z, Z̄]]] and Im[Z̄, [Z, [Z, Z̄]]] are linearly dependent on the vectors ReZ, ImZ and Im[Z, Z̄] at the point F (s(p)).

The following lemma (whose proof is based on direct calculations) is the main step in the proof of the proposition.

Lemma 4. The set N is a stratified (in the sense of Whitney, see [GWPL], p.11) subset of J^{4}(K, N). Moreover, for each point p ∈ K the corresponding fiber N_{p} of N has real codimension 5 in the fiber of J^{4}(K, N) over p.

Proof of lemma 4. Let p be an arbitrary point of K, suppose s is a germ of a smooth section of N defined in a neighbourhood of p and s(p) is the point of s that corresponds to p under the natural projection from N to K. Consider a linear change of coordinates in ℂ^{3} such that the point F (s(p)) is the origin in the new coordinates z_{1}, z_{2}, w(w = u + iv) and such that in a neighbourhood of this point one has

G = {v > H(z_{1}, z_{2}, u)},

where H is a C^{∞}-smooth function with the properties

H(0) = H′_{z_{1}}(0) = H′ ; z̄_{1}(0) = H′_{z_{2}}(0) = H′ ; z̄_{2}(0) = H′_{u}(0) = 0.  (1)

If we choose the z_{1}-axis to be equal to T_{0}^{ℂ}(K_{s}), then, maybe after replacing z_{2} by a suitable linear function of z_{2} and w, one has in a neighbourhood of the origin that

K_{s} = {u = Ψ(z_{1}, z_{2}), v = H(z_{1}, z_{2}, u)},

where Ψ is a C^{∞}-smooth function defined in a neighbourhood of the origin in ℂ_{z_{1},z_{2}}^{2} such that

Ψ(0) = Ψ′_{z_{1}}(0) = Ψ′ ; z̄_{1}(0) = 0, Ψ′_{z_{2}}(0) = Ψ′ ; z̄_{2}(0) = 1.  (2)

Direct calculation shows that the following (1,0)-vector field Z defined in a neighbourhood of the origin is complex tangential to K_{s}: Z = ∂_{z_{1}} + a∂_{z_{2}} + b∂_{w}, where

a = −\frac{H′_{z_{1}} + (H′_{u} + i)Ψ′_{z_{1}}}{H′_{z_{2}} + (H′_{u} + i)Ψ′_{z_{2}}}  (3)

and

b = −2\frac{H′_{z_{1}}Ψ′_{z_{2}} − H′_{z_{2}}Ψ′_{z_{1}}}{H′_{z_{2}} + (H′_{u} + i)Ψ′_{z_{2}}} .  (4)

Note, that any vector field which is complex tangential to K_{s} is uniquely determined by its projection onto the (z_{1}, z_{2})-plane. Therefore, we can consider the vector field Z^{∗} = ∂_{z_{1}} + a∂_{z_{2}} instead of Z. Then the computation below of the Lie brackets of Z^{∗} and Z̄^{∗} will give us the (z_{1}, z_{2})-components of the Lie brackets of Z and Z̄ in the jet space J^{4}(K, N). For the first Lie bracket [Z^{∗}, Z̄^{∗}] we obtain

[Z^{∗}, Z̄^{∗}] = −2iIm(Z̄^{∗}(a)∂_{z_{2}}) = −2iIm(a′ ; z̄_{1}∂_{z_{2}} + a′ ; z̄_{2} ā∂_{z_{2}}).  (5)

To compute the coefficients explicitly, put

a = α · A

where α is the numerator and −\frac{1}{A} the denominator of the fraction in (3). Then by (1) and (2)

α(0) = 0, A(0) = i.

Direct computation gives

a′ ; z̄_{1} = α′ ; z̄_{1}A + αA′ ; z̄_{1}
a′ ; z̄_{2} = α′ ; z̄_{2}A + αA′ ; z̄_{2}
Z̄^{∗}(a) = a′ ; z̄_{1} + a′ ; z̄_{2} ā = α′ ; z̄_{1}A + αA′ ; z̄_{1} + α′ ; z̄_{2} · ᾱ · |A|^{2} + |α|^{2}A′ ; z̄_{2} Ā.  (6)

Here

α′ ; z̄_{1} = H′′_{z̄_{1}z_{1}} + H′′_{z̄_{1}u}Ψ′_{z_{1}} + (H′_{u} + i)Ψ′′_{z̄_{1}z_{1}} = H′′_{z̄_{1}z_{1}}(0) + iΨ′′_{z̄_{1}z_{1}}(0) + O(z).

All terms in (6) except the first one are of order O(z), hence

Z̄^{∗}(a) = iH_{z̄_{1}z_{1}}(0) − Ψ′′_{z̄_{1}z_{1}}(0) + O(z).

It follows from (5) that

\begin{aligned}[Z^{∗}, Z̄^{∗}] = −2iIm(Z̄^{∗}(a)∂_{z_{2}}) & = −i(H′′_{z̄_{1}z_{1}}(0)∂_{x_{2}} + Ψ′′_{z̄_{1}z_{1}}(0)∂_{y_{2}})+ \\ & O(z)∂_{x_{2}} + O(z)∂_{y_{2}}. \end{aligned}  (7)

On the next step we compute the second Lie bracket and extract the derivatives of the highest (third) order

[Z^{∗}, [Z^{∗}, Z̄^{∗}]] = [∂_{z_{1}} + a∂_{z_{2}}, Z^{∗}(ā)∂z̄_{2} − Z̄^{∗}(a)∂_{z_{2}}] = (∂_{z_{1}}Z^{∗}(ā))∂z̄_{2}−
− (∂_{z_{1}} Z̄^{∗}(a))∂_{z_{2}} + a(∂_{z_{2}}Z^{∗}(ā))∂z̄_{2} − a(∂_{z_{2}} Z̄^{∗}(a))∂_{z_{2}} − Z^{∗}(ā)(∂z̄_{2}a)∂_{z_{2}}+
+ Z̄^{∗}(a)(∂_{z_{2}}a)∂_{z_{2}}.

Since α(0) = 0 and since the last two terms in the preceding expression contain only derivatives of H and Ψ up to second order, it follows from (1), (2) and (6) that

\begin{aligned}[Z^{∗}, [Z^{∗}, Z̄^{∗}]] & = −i(H′′′_{z_{1}z_{1}z̄_{1}}(0)∂_{x_{2}} + Ψ′′′_{z_{1}z_{1}z̄_{1}}(0)∂_{y_{2}}) + (A_{2}∂_{x_{2}} + B_{2}∂_{y_{2}})+ \\ & + O(z)∂_{x_{2}} + O(z)∂_{y_{2}}, \end{aligned} (8)

where A_{2} and B_{2} consist of terms containing only first and second order derivatives of H and Ψ.

Similarly,

\begin{aligned}[Z^{∗}, [Z^{∗}, [Z^{∗}, Z̄^{∗}]]] & = −i(H_{z_{1}z_{1}z_{1}z̄_{1}}^{(4)}(0)∂_{x_{2}} + Ψ_{z_{1}z_{1}z_{1}z̄_{1}}^{(4)}(0)∂_{y_{2}})+ \\ & + (A_{3}∂_{x_{2}} + B_{3}∂_{y_{2}}) + O(z)∂_{x_{2}} + O(z)∂_{y_{2}} \end{aligned}  (9)

and

\begin{aligned}[Z̄^{∗}, [Z^{∗}, [Z^{∗}, Z̄^{∗}]]] & = −i(H_{z_{1}z_{1}z̄_{1}z̄_{1}}^{(4)}(0)∂_{x_{2}} + Ψ_{z_{1}z_{1}z̄_{1}z̄_{1}}^{(4)}(0)∂_{y_{2}})+ \\ & + (A′_{3}∂_{x_{2}} + B′_{3}∂_{y_{2}}) + O(z)∂_{x_{2}} + O(z)∂_{y_{2}}, \end{aligned}  (10)

where A_{3}, B_{3}, A′_{3} and B′_{3} consist of terms containing only derivatives of H and Ψ up to the third order.

To finish the proof of lemma 4, let us first note that the real vector fields U_{1} = ReZ and V_{1} = ImZ are complex tangential to K_{s} and that the Lie brackets up to order 3 of these vector fields are equal (up to multiplication by non-zero constants) to the vector fields V_{2} = Im[Z, Z̄], U_{3} = Re[Z, [Z, Z̄]], V_{3} = Im[Z, [Z, Z̄]] and linear combinations of the vector fields U_{4} = Re[Z, [Z, [Z, Z̄]]], V_{4} = Im[Z, [Z, [Z, Z̄]]] and W_{4} = Im[Z̄, [Z, [Z, Z̄]]]. Since, by our assumptions, the domain G is strictly pseudoconvex, it follows that H′′_{z̄_{1}z_{1}}(0) > 0. Hence (7) implies that the vector fields U_{1}, V_{1} and V_{2} are linearly independent. The subset N of the jet space J^{4}(K, N) is defined by the condition that each of the vectors U_{3}, V_{3}, U_{4}, V_{4} and W_{4} is a linear combination of U_{1}, V_{1}, V_{2}. The property of a vector to be a linear combination of some other vectors is an algebraic relation on the coordinates of the vectors. Formulas (3)-(10) above show that the coordinates of the vector fields U_{1}, V_{1}, V_{2}, U_{3}, V_{3}, U_{4}, V_{4} and W_{4} are rational functions of the derivatives of the functions H and Ψ. We conclude that N is defined in J^{4}(K, N) by algebraic equations and therefore is a stratified subset of J^{4}(K, N). Moreover, from formulas (3)–(10) one obtains 5 equations, each of them describing the condition for one of the vectors U_{3}, V_{3}, U_{4}, V_{4} and W_{4} to be a linear combination of the vectors U_{1}, V_{1} and V_{2}. The mentioned formulas show that in different equations the appearing partial derivatives of H and Ψ of highest order are different. Hence, the 5 equations define N as a complete intersection and the codimension of N in J^{4}(K, N) is equal to 5. The proof of lemma 4 is completed. □

To finish the proof of the proposition, let us consider a locally finite covering {B_{n}}_{n=1}^{∞} of K by balls (in the intrinsic metric of K) such that the radius of every ball B_{n} is less than half of the distance (also in the intrinsic metric of K) from the center of the ball to the boundary ∂K of K. Consider a sequence of balls {B′_{n}}_{n=1}^{∞} in K, which is also a covering of K, such that B′_{n} ⊂⊂ B_{n} for all n. Since dim_{ℝ} K = 4, it follows from our lemma 4 and theorem 1.1 on p.127 in [GWPL], that there is a perturbation K_{1} of K in C^{∞} which keeps K unchanged outside B_{1} and such that the 4-jet of K_{1} over B̄′_{1} does not intersect the set N. Then, again by our lemma 4 and theorem 1.1 in [GWPL], there is a perturbation K_{2} of K_{1} in C^{∞} which keeps K_{1} unchanged outside B_{2} and such that the 4-jet of K_{2} over B̄′_{2} does not intersect the set N. Moreover, this perturbation of K_{1} can be chosen so small in C^{4}-metric that the 4-jet of K_{2} is still disjoint from N over B̄′_{1}. In a similar way we can inductively construct a perturbation K_{3} of K_{2}, K_{4} of K_{3} and so on. On the step number m our construction looks as follows. Let K_{m} be a perturbation of K_{m−1} in C^{∞} (guaranteed by lemma 4 and theorem 1.1 of [GWPL]) which keeps K_{m−1} unchanged outside B_{m} and such that the 4-jet of K_{m} over B̄′_{m} is disjoint from N. We required on step m−1 that the 4-jet of K_{m−1} is disjoint from N over ∪_{i=1}^{m−1}B̄′_{i}. On step m we choose K_{m} close enough to K_{m−1} so that the 4-jet of K_{m} is still disjoint from N over ∪_{i=1}^{m−1}B̄′_{i} (and, therefore, over ∪_{i=1}^{m}B̄′_{i}). Since the covering {B_{n}}_{n=1}^{∞} was chosen to be locally finite, it follows that for each n the pertubations K_{m} will stay the same over the ball B_{n} for all m big enough. Therefore, we have a well defined limit K^{∗} of K_{m} as m → ∞ which is C^{∞}-smooth and, by our construction, the 4-jet of K^{∗} is disjoint from the set N over the whole of ∪_{m=1}^{∞}B̄′_{m} = K. Then, by the definition of the set N, we have that the set K^{∗} is of type 4 and, hence, is globally minimal. Moreover, since the balls of our covering are getting smaller when we approach the boundary ∂K of K, it follows that if we choose perturbations K_{m} is such a way, that the distance from the 4-jet of K_{m} to the 4-jet of K_{m−1} is decreasing to zero rapidly enough as m → ∞, then the manifold M̃ = (M \ K) ∪ K^{∗} will still be C^{4}-smooth. This completes the proof of proposition. □

Remark 4. Proposition is proved under the assumption that the domain G has C^{∞}-smooth boundary. In this case the set K can be perturbed to a C^{∞}-smooth set K^{∗} of type 4. One can easily adapt our arguments to the case of domains with C^{4}-smooth boundary. Then the perturbed set K^{∗} will also be C^{4}-smooth.

Theorem 1 can be generalized in the following way.

Theorem 2. Let M be an orientable smooth manifold of real dimension 4 contained in the unit sphere ∂𝔹^{3}. Suppose M has smooth boundary ∂M which is the disjoint union of finitely many connected smooth compact maximally complex CR-manifolds of dimension 3. Then there exist arbitrarily small smooth deformations M̃ of M with fixed boundary ∂M̃ = ∂M such that M̃ has only finitely many complex tangencies q_{1}, …, q_{N} and the manifold M̃ \ {q_{1}, …, q_{N}} is globally minimal.

Moreover, the essential envelope of holomorphy E(∂𝔹^{n}\ M^{≂}) is multi-sheeted. Each point of M^{≂} is in the closure of points over which E(∂𝔹\ M^{≂}) is multi-sheeted.

Proof. Let N be one of the boundary components of M. Then a small neighbourhood of N on M is a generic CR-manifold (although the whole manifold M may not be so). Indeed, by Forstnerič [F], for each p ∈ N the tangent space T_{p}N is not contained in the complex tangent space T_{p}^{c}∂𝔹^{3}. Hence, for p ∈ N the tangent space T_{p}M̄ is not contained in T_{p}^{c}∂𝔹^{3}. By continuity this is true for all points p′ ∈ M close to N.

It follows from the jet transversality theorem (see e.g. [Hi] Ch. 3) that there is a small perturbation M^{∗} of M which fixes M near ∂M such that the tangent space to M^{∗} is not complex at all points of M^{∗}, except maybe for a finitely many points {q_{1}, q_{2}, …, q_{N}}. Indeed, consider for each point p ∈ ∂𝔹^{3} the set of all oriented 4-planes T_{p}^{4} such that p ∈ T_{p}^{4} ⊂ T_{p}(∂𝔹^{3}). Each plane from this family is uniquely determined by the unit vector n_{p} ∈ T_{p}(∂𝔹^{3}) which is ortogonal to T_{p}^{4} and such that the pair (T_{p}^{4}, n_{p}) defines the natural orientation on T_{p}(∂𝔹^{3}). Therefore, this family of 4-planes is diffeomorphic to the space of unit vectors in a 5-dimensional euclidean space, i.e. to S^{4}. Among these 4-planes there are two planes that correspond to the complex tangent space T_{p}^{ℂ}(∂𝔹^{3}) (one with positive orientation, the other one with negative). We obtain a set consisting of 2 points, i.e. a submanifold of S^{4} which has codimension 4. This means that the jet transversality theorem is used to approximate the inclusion map of the manifold M into ∂𝔹^{3} by a map whose 1-jet map is transversal to a codimension 4 subset of the 1-jet space. Since the dimension of M is also 4 and M is proved to be without complex tangencies near ∂M, we obtain the required M^{∗}.

The proposition above gives a small perturbation M̃ of M^{∗} which fixes ∂M and the complex tangencies such that M̃ \ {q_{1}, …, q_{N}} is a globally minimal generic CR-manifold contained in ∂𝔹^{n}. This situation is enough for the proof of theorem 1 to go through. □

Now we come back to the examples and prove that in the mentioned situations the envelope of holomorphy is exactly two-sheeted.

Claim 1. Let M be as in example 1. If M^{≂} is chosen as described there and close in C^{4} norm to M̄ then E(∂𝔹^{3}\ M^{≂}) is two-sheeted.

Claim 2. Let c ∈ (0, \frac{1}{\sqrt{3}^{3}} ) and close to \frac{1}{\sqrt{3}^{3}} . Denote by M the set M = \{z = (z_{1}, z_{2}, z_{3}) ∈ ∂𝔹^{3} : z_{1} · z_{2} · z_{3} ∈ (c, +∞)}. Then M is a smooth 4-dimensional manifold. Its boundary ∂M = N is a connected compact maximally complex CRmanifold. There exist arbitrarily small (in C^{4}) smooth deformations M^{≂} of M̄ with ∂M̃ = ∂M such that M̃ is a generic globally minimal CR-manifold. In this case E(∂𝔹^{3}\ M^{≂}) is two-sheeted. The whole set M^{≂} is contained in the closure of the set of points over which E(∂𝔹^{3}\ M^{≂}) is two-sheeted.

We will prove here claim 2. Claim 1 is proved in the same way but simpler in view of the simpler topology of the first example.

Proof of Claim 2. The set

H = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : Im(z_{1} · z_{2} · z_{3}) = 0, Re(z_{1} · z_{2} · z_{3}) > c}

is a smooth (even real analytic) hypersurface which is foliated by the complex analytic hypersurfaces H_{α} = {z_{1} ·z_{2} ·z_{3} = α}, α ∈ (c, +∞). If a point is contained in H, then none of its coordinates vanishes. It is easy to see that H intersects ∂𝔹^{3} transversally, so M is a smooth 4-dimensional manifold. M is foliated by the 3-dimensional compact maximally complex CR-manifolds N_{α} = {z_{1} · z_{2} · z_{3} = α\} ∩ ∂𝔹^{3}, α ∈ (c, \frac{1}{\sqrt{3}^{3}} ). For α = \frac{1}{\sqrt{3}^{3}} the intersection H_{α} ∩ ∂𝔹^{3} degenerates. It is a 2-torus which consists of complex tangencies of M. For α > \frac{1}{\sqrt{3}^{3}} , H_{α} does not meet ∂𝔹^{3}. It is clear that ∂M = N(= H_{c} ∩ ∂𝔹^{3}).

Choose a generic CR-manifold M^{∗} ⊂ ∂𝔹^{3} such that \overline{M^{∗}} is a C^{4} small perturbation of M̄ with ∂M^{∗} = ∂M (see [JSh]). Let M^{≂} be C^{4} close to \overline{M^{∗}} with ∂M̃ = ∂M, such that M̃ is globally minimal (see proposition).

It will be convenient to use the following coordinates

(z_{1}, z_{2}, ζ_{3}) = (z_{1}, z_{2}, z_{1}z_{2}z_{3})

on the set

Ω = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : Re(z_{1}z_{2}z_{3}) > c}.

Ω is a domain of holomorphy. In these coordinates M has the form

M = \{(z_{1}, z_{2}, ξ_{3} + iη_{3}) : η_{3} = 0, ξ_{3} > c, |z_{1}|^{2} + |z_{2}|^{2} + \frac{ξ_{3}^{2}}{|z_{1}|^{2}|z_{2}|^{2}} = 1\}.

The manifold N = ∂M bounds the analytic manifold V = 𝔹^{3} ∩ H_{c}. The set M ∪ N ∪ V is a closed piecewise smooth manifold. It bounds the following set Q = H ∩ 𝔹^{3} ⊂ H,

\begin{aligned}Q & = \{(z_{1}, z_{2}, ξ_{3} + iη_{3}) ∈ Ω ∩ 𝔹^{3} : η_{3} = 0\} ≅ \\ & ≅ \{(z_{1}, z_{2}, ξ_{3}) ∈ ℂ^{2} × ℝ : ξ_{3} > c, |z_{1}|^{2} + |z_{2}|^{2} + \frac{ξ_{3}^{2}}{|z_{1}|^{2}|z_{2}|^{2}} < 1\}. \end{aligned}

The manifold M̃ has the form

M̃ = {(z_{1}, z_{2}, ξ_{3} + iη_{3}) : η_{3} = E_{3}(z_{1}, z_{2}, ξ_{3}), (z_{1}, z_{2}, ξ_{3}) ∈ Γ},

where

Γ = \{(z_{1}, z_{2}, ξ_{3}) : ξ_{3} > c, |z_{1}|^{2} + |z_{2}|^{2} + \frac{ξ_{3}^{2}}{|z_{1}|^{2}|z_{2}|^{2}} = 1 − E_{3}^{2}(z_{1}, z_{2}, ξ_{3})\}

and E_{3} is a small smooth real valued function. We consider Γ as a subset of H and M̃ as a graph over Γ. Note that Γ ⊂ Q̄. (Points (z_{1}, z_{2}, ξ_{3}) ∈ Γ for which E_{3} = 0 are contained in M ⊂ ∂Q, points for which E_{3} ≠ 0 are contained in Q.) Moreover, ∂Γ = ∂M and Γ ∪ N ∪ V bounds an open subset G of Q.

Our plan is now the following. Let U be a one-sided neighbourhood of ∂𝔹^{3}\M̃ with a priori analytic extension. Note first that each analytic function u in U extends to a neighbourhood of 𝔹^{3} \ (Ω ∪ V) (i.e. to the union of 𝔹^{3} \ Ω̄ with a neighbourhood of (𝔹^{3} ∩ ∂Ω) \ V).

In fact for c′ + it ∈ ℂ, |c′ + it| < \frac{1}{\sqrt{3}^{3}} , the analytic hypersurface H_{c′+it} = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : z_{1}z_{2}z_{3} = c′ + it} intersects ∂𝔹^{3} transversally. For those c′ + it the intersection H_{c′+it} ∩ 𝔹^{3} is a Stein manifold. If moreover, c′ < c, or t ≠ 0 and |c − c′| is small compared with |t|, then the boundary of H_{c′+it} ∩ 𝔹^{3} is contained in ∂𝔹^{3}\ M^{≂}. Hence, for such c′ + it each analytic function u in U extends to a neighbourhood of H_{c′+it} ∩ 𝔹^{3}. The assertion follows by applying the Hartogs-Bochner theorem to 𝔹^{3} ∩ {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : Re(z_{1}z_{2}z_{3}) < c − ε} for some small ε > 0. (The function u is analytic near 𝔹^{3} ∩ {Re(z_{1}z_{2}z_{3}) = c − ε} and in a one-sided neighbourhood of ∂𝔹^{3} ∩ {Re(z_{1}z_{2}z_{3}) ≤ c − ε}.)

Note that M̃ divides ∂𝔹^{3} ∩ Ω into two connected components ω_{+} and ω_{−}. Let U_{+} and U_{−} be one-sided neighbourhoods of ω_{+} and ω_{−}, respectively, with a priori analytic extension to U_{+} and U_{−}, respectively. Assume ω_{+} is on the side of M̃ where η_{3} > E_{3}(z_{1}, z_{2}, ξ_{3}).

Denote by C(G) the “cylinder over G”,

C(G) = {(z_{1}, z_{2}, ζ_{3}) = (z_{1}, z_{2}, ξ_{3} + iη_{3}) : η_{3} ∈ ℝ, ξ_{3} ∈ G}.

Claim 3. C(G) is pseudoconvex, provided the constant c in claim 2 is close to \frac{1}{\sqrt{3}^{3}} and M̃ is close to M.

Proof of claim 3. Denote by C(Q) the cylinder over Q. Both domains, C(G) and C(Q), are invariant under translation in the η_{3}-direction and the projection parallel to the η_{3}-direction is equal to G and Q, respectively. It is enough to prove that the boundary points of C(Q), for which ξ_{3} = \frac{1}{\sqrt{3}^{3}} , are strictly pseudoconvex boundary points for C(Q).

Indeed, if c is close to \frac{1}{\sqrt{3}^{3}} , then for all points of ∂C(Q) which project to M ⊂ ∂Q the ξ_{3}-coordinate is close to \frac{1}{\sqrt{3}^{3}} and hence the corresponding point is a strictly pseudoconvex boundary point of C(Q). Since the boundary part Γ of G is C^{2} close to M, all points of ∂C(G) which project to Γ are strictly pseudoconvex boundary points. The remaining boundary points of ∂C(G) are contained in the real hyperplane {ξ_{3} = c} (in coordinates (z_{1}, z_{2}, ζ_{3})) and hence they are pseudoconvex boundary points as well.

Prove now that all points of the set

E = \{(z_{1}, z_{2}, ξ_{3}) ∈ ∂C(Q), ξ_{3} = \frac{1}{\sqrt{3}^{3}} \}

(and hence all their translates in η_{3}-direction) are strictly pseudoconvex boundary points. In (z_{1}, z_{2}, z_{3})-coordinates E consists of those points (z_{1}, z_{2}, z_{3}) ∈ ∂𝔹^{3} for which |z_{1}| = |z_{2}| = |z_{3}| = \frac{1}{\sqrt{3}}. The analytic hypersurface \{z_{1} · z_{2} · z_{3} = \frac{1}{\sqrt{3}^{3}} \} is tangent to ∂𝔹^{3} at those points (see above or [JSh] for more details). In (z_{1}, z_{2}, ζ_{3})- coordinates this means that the complex hyperplane \{ζ_{3} = \frac{1}{\sqrt{3}^{3}} \} is tangent to ∂𝔹^{3} at points of E. Hence ∂𝔹^{3} and ∂C(Q) have the same complex tangent space at points p ∈ E (we use (z_{1}, z_{2}, ζ_{3})-coordinates).

Each point p ∈ E is strictly pseudoconvex for ∂𝔹^{3} (in (z_{1}, z_{2}, ζ_{3}) coordinates). Since (near p ∈ E) ∂𝔹^{3} and ∂C(Q) do not differ in complex tangent directions (their intersections with {η_{3} = 0} coincide near p ∈ E) the Levi form of ∂C(Q) at p ∈ E is strictly positive in complex tangent directions. □

Claim 4. There exist two continuous functions h_{+} and h_{−} on Ḡ such that 1. h_{+} ≥ h_{−} on G.

2. h_{+} = h_{−} = E_{3} on Γ and h_{+} = h_{−} = 0 on N ∪ V. (i.e. the graphs g_{+}(Ḡ) and g_{−}(Ḡ) of the functions h_{+} and h_{−} over Ḡ, g_{±}(Ḡ) = {(z_{1}, z_{2}, ξ_{3} + ih_{±}(z_{1}, z_{2}, ξ_{3}) : (z_{1}, z_{2}, ξ_{3}) ∈ Ḡ}, have common boundary which is equal to the union of V̄ and M̃, M̃ being equal to the graph of E_{3} over Γ.)

3. The function u extends analytically to the relatively compact domain bounded by ω_{+}, by part of ∂Ω and by the graph g_{+}(Ḡ) of h_{+} over Ḡ. (Denote this domain B_{+}.)

Similarly, u extends analytically to the domain bounded by ω_{−}, by part of ∂Ω and by g_{−}(Ḡ). (Denote this domain B_{−}.)

Suppose for a moment that claim 4 is proved. We will use the following result due to Trepreau [Tr1], see also [Č2].

Let C_{+} be the “semi-cylinder” over G

C_{+} = {(z_{1}, z_{2}, ξ_{3} + iη_{3}) : (z_{1}, z_{2}, ξ_{3}) ∈ G, η_{3} > h_{+}}.

Its envelope of holomorphy equals C_{+}^{∗} which is a semi-cylinder of the same form with h_{+} replaced by h_{+}^{∗}(h_{+}^{∗} ≤ h_{+}).

To apply the result, assume (changing h_{+} slightly, if necessary) that u is analytic in B_{+} and in a neighbourhood of the graph of h_{+} over G. By an argument of Trepreau (see e.g. [Tr2]), u can be written as u = u_{+}+u_{−}, where u_{+} is analytic in C_{+} and on the graph of h_{+} and u_{−} is analytic in C(G) \ C̄_{+} and on the graph of h_{+}. (The idea is to multiply u by a smooth cut-off function in C_{+} which is 1 near the graph of h_{+} and extend by zero to get a smooth function v in C_{+}. Let (¯∂v)_{+} be equal to ¯∂v in C_{+} and zero else. Since C(G) is pseudoconvex, there is a function w in C(G) such that ¯∂w = (¯∂v)_{+}. Put u_{+} = v − w, u_{−} = w.) Hence, u_{+} extends to C_{+}^{∗}, and, therefore, u = u_{+} + u_{−} extends analytically to C_{+}^{∗} ∩ 𝔹^{3}.

In the same way, u extends analytically to C_{−}^{∗} ∩ 𝔹^{3} with C_{−}^{∗} defined symmetrically. We obtained that u has analytic continuation to the following domains: a neighbourhood of 𝔹^{3} \(Ω∪V), B_{+}, C_{+}^{∗} and B_{−}, C_{−}^{∗}. (Note that B_{+} \C_{+}^{∗} consists of points (z_{1}, z_{2}, ζ_{3}) ∈ 𝔹^{3} for which (z_{1}, z_{2}, ξ_{3}) ∈ Q \ G). Continuation of u coincides on the intersection of any pair of the domains above, except, maybe, C_{+}^{∗} and C_{−}^{∗}, which contain the overlapping wedges W^{+} and W^{−}, respectively. The domains above cover a two-sheeted Riemann domain over ℂ^{3} (all pairs of domains are glued along their intersection except C_{+}^{∗} and C_{−}^{∗}). Denote by π the natural projection to ℂ^{3} defined on the suitably defined closure of the Riemann domain (see [O] or [GR]). Each boundary point of the Riemann domain is pseudoconvex. Indeed, this is clear for points which project to ∂𝔹^{3}, it is also clear for points which project to the graph of either h_{+}^{∗} over the open set G or of h_{−}^{∗} over G. The remaining boundary part of the Riemann domain projects homeomorphically to the analytic hypersurface V. Note that there are points in ℂ^{3} close to V which are covered twice by the Riemann domain. Changing coordinates in a neighbourhood of π^{−1}(V) on the Riemann domain (e.g. by lifting (z_{1}, z_{2}, \sqrt{ζ_{3} − c}) to the Riemann domain) one can easily see that π^{−1}(V) consists of pseudoconvex points (lying on a 2-dimensional analytic manifold contained in the boundary).

By theorems of Oka and Grauert, the Riemann domain is Stein (see [G], [O]). Hence it coincides with the envelope of holomorphy E(∂𝔹^{3}\ M^{≂}).

It remains to give the

Proof of claim 4. Let u be an analytic function in the one-sided neighbourhood U of ∂𝔹^{3} \M̃ with a priori analytic extension. Note, that parts of ω_{+} and ω_{−} (the connected components of ∂𝔹^{3} \ M̃) can be represented as graphs over G. Recall that

G = \{(z_{1}, z_{2}, ξ_{3}) : ξ_{3} > c, |z_{1}|^{2} + |z_{2}|^{2} + \frac{ξ_{3}^{2}}{|z_{1}|^{2}|z_{2}|^{2}} < 1 − E_{3}^{2}(z_{1}, z_{2}, ξ_{3})\}.

For (z_{1}, z_{2}, ξ_{3}) ∈ Ḡ we put |E_{3}(z_{1}, z_{2}, ξ_{3})| = (1 − |z_{1}|^{2} − |z_{2}|^{2} − \frac{ξ_{3}^{2}}{|z_{1}|^{2}|z_{2}|^{2}} )^{1/2}. This function is a continuous extension of |E_{3}| from the part of ∂G where it was defined previously. Then

∂𝔹^{3} ∩ C(G) = {(z_{1}, z_{2}, ξ_{3} + iη_{3}) : (z_{1}, z_{2}, ξ_{3}) ∈ G, η_{3} = ±|E_{3}(z_{1}, z_{2}, ξ_{3}|}.

The “upper” and “lower” component of ∂𝔹^{3}∩C(G), which correspond respectively to the sign + and to the sign − in the last formula, are graphs of functions over G and both components are contained in ∂𝔹^{3} \ M̃ (more precisely, in ω_{+} and ω_{−} respectively).

The two functions |E_{3}| and −|E_{3}| do not coincide at points p ∈ ∂G where E_{3} ≠ 0 (i.e., at points of ∂G which are contained in Q). In this case 𝔹^{3} ∩ ∂C(G) containes a segment of the form {p + i(0, 0, η_{3}) : η_{3} ∈ (−|E_{3}|, |E_{3}|)}.

Our aim is to prove that u extends analytically to all these segments. More precisely, we want to prove the following. Let ∂_{j}^{+}(G) be the components of ∂G where E_{3} is negative and ∂_{j}^{−}(G), respectively, the components where E_{3} is positive. Denote by ∂_{j}^{+} and ∂_{j}^{−} the corresponding components of 𝔹^{3} ∩ ∂C(G) and by A_{j}^{+} and A_{j}^{−} the components of 𝔹^{3} \\overline{C(G)} which are bounded by ∂_{j}^{+} (∂_{j}^{−}, respectively) and a subset of ∂𝔹^{3}. Note, that the following part of the boundary of ∂_{j}^{+}

{p + i(0, 0, η_{3}) : p ∈ ∂_{j}^{+}(G), η_{3} = E_{3}(p) < 0}

is contained in M̃, while the part

{p + i(0, 0, η_{3}) : p ∈ ∂_{j}^{+}(G), η_{3} = |E_{3}(p)| > 0}

is contained in ∂𝔹^{3} \ M̃, hence in ω^{+}. We assert that the function u (in fact, the function u_{+}), which was originally defined in a one-sided neighbourhood of ω_{+}, extends to an analytic function in ∪_{j}A_{j}^{+} and on a neighbourhood of ∪_{j}∂_{j}^{+}. The analogous assertion holds for u_{−}.

Assume for a moment that this assertion is true and prove that this implies claim 4. In fact, join the “upper” component of ∂𝔹^{3} ∩ C(G) with ∪_{j}∂_{j}^{+} and with the union of all common “edges”

{p + i(0, 0, η_{3}) : p ∈ ∂_{j}^{+}(G); η_{3} = |E_{3}(p)| > 0} ⊂ ω_{+}.

Join to this set a part of the hypersurface which bounds Ω, namely, the set \{(z_{1}, z_{2}, ζ_{3}) ∈ \overline{𝔹^{3}} : ξ_{3} = c, η_{3} > 0\}. The union is a manifold with boundary M̃ ∪ N ∪ V. The function u_{+} is analytic in a one-sided neighbourhood of the upper part of ∂𝔹^{3} ∩ C(G) and by our assumption it extends analytically to the rest of the manifold and to ∪_{j}A_{j}^{+}.

Move all points of the manifold slightly to the side of 𝔹^{3} ∩ C(G) (fixing the boundary) so that we obtain a graph of a continuous function h_{+} over Ḡ, the boundary of the graph being M̃ ∪N ∪V, and the function u_{+} extends analytically to the domain B_{+} bounded by ω_{+}, by part of ∂Ω and by the graph of h_{+} over Ḡ.

The analogous construction gives the function h_{−} on Ḡ. It is clear from the construction that h_{+} > h_{−} on G and h_{+} = h_{−} on ∂G.

Prove now the mentioned assertion that the function u_{+} has analytic extension to ∪_{j}A_{j}^{+} and to a neighbourhood of ∪_{j}∂_{j}^{+}. We will use here the fact that ∂C(Q) is sufficiently close to ∂C(G) (e.g. measured in C^{2} metric for ∂Q and ∂G which form the bases for the cylinders ∂C(Q) and ∂C(G)), and C(Q) is strictly pseudoconvex. The latter implies that for each point p ∈ ∂C(Q) there exists a ball b_{p} (in ℂ^{3}) with center p and a biholomorphic change of coordinates in b_{p} which makes b_{p} ∩∂C(Q) strictly convex from the side of C(Q). If ∂C(G) is close enough to ∂C(Q) the same is true for each point p ∈ ∂C(G) and a compactness argument (for the bases ∂Q and ∂G) allows to take the radius of the balls b_{p} estimated from below by a constant r > 0 depending only on Q. Moreover, the Hessian of the defining function of b_{p} ∩∂C(G) in the mentioned coordinates has a uniform estimate. This implies in particular, the following. If we denote by h_{p} the tangent space T_{p}∂C(G) to ∂C(G) at p in the chosen coordinates then ∂C(G) ∩ b_{p} \ {p} is on one side of h_{p} and, moreover, ∂C(G) ∩ ∂b_{p} has distance ≥ ar^{2} from h_{p} ∩ ∂b_{p} for some positive constant a depending only on Q.

Let p ∈ ∂_{j}^{+} for some j (hence, p ∈ C(G) ∩ 𝔹^{3} ⊂ ∂C(G) \ ∂C(Q)). If the C^{2} distance of ∂C(Q) and ∂C(G) is small compared to the value ar^{2}, then ∂C(Q)∩∂b_{p} is closer to ∂C(G) ∩ ∂b_{p} than to h_{p} ∩ ∂b_{p} and hence ∂C(Q) ∩ ∂b_{p} is on the same side of h_{p} as ∂C(G) ∩ ∂b_{p}.

Denote by ⊓_{p} the closed halfspace (in b_{p}) which corresponds to h_{p} and does not contain points of ∂C(G) \ {p}. The just mentioned arguments imply that ∂C(Q) ∩ ⊓_{p} is compact (in other words ∂C(Q) ∩ ⊓_{p} is separated away from ∂b_{p}). Moreover, no point in ∂C(Q) ∩ ⊓_{p} is contained in ∂C(G) (since ∂C(G) \ {p} ⊂ b_{p} \ ⊓_{p} and p ∉ ∂C(Q)).

Note that by the definition of Q the inclusion Ω ∩ \overline{𝔹^{3}} ⊂ \overline{C(Q)} holds, hence ∂𝔹^{3} \ \overline{C(G)} ⊂ \overline{C(Q)} \ \overline{C(G)}. It follows that ∂𝔹^{3} ∩ ⊓_{p} is compact (i.e. separated away from ∂b_{p}). Applying e.g. Kontinuitätssatz we obtain analytic continuation u_{p} of u to a neighbourhood of 𝔹^{3} ∩⊓_{p}. Replace ⊓_{p} by a relatively closed subset of ⊓_{p} which is convex (in Euclidean coordinates) and contains the point p. E.g. we may take b_{p} ∩ K_{p}, where K_{p} is a closed convex cone with vertex p and symmetry axis being the outer normal to C(G) at p.

Each point which is close to the strictly pseudoconvex part of ∂C(G) and not contained in C(G) is contained in b_{p} ∩ K_{p} for some p. In particular, this holds for points in 𝔹^{3} \ C(G).

We need to show that if for some p and p′ in a connected component A_{j}^{+} ∪ ∂_{j}^{+} the convex (and hence connected) set b_{p} ∩ K_{p} ∩ K_{p′} ∩ 𝔹^{3} is non-empty, then u′_{p} = u_{p} on this set. This holds if p and p′ are close, since then the set contains points in 𝔹^{3} which are close to ∂𝔹^{3}. A connectedness argument gives the required equality for all p, p′ ∈ A_{j}^{+} ∪ ∂_{j}^{+}. We obtained a well-defined analytic function in a neighbourhood of A_{j}^{+} ∪ ∂_{j}^{+}. □

Note that here we gave an alternative proof of the fact that E(∂𝔹^{3}\ M^{≂}) is multi-sheeted. It avoided the construction of a holomorphic function in the one-sided neighbourhood U which separates W^{+} and W^{−}. Instead we found a pseudoconvex Riemann domain over ℂ^{3} which contains U and non-intersecting copies of W^{+} and W^{−} with edges m^{+} and m^{−} glued to U. We are not able to construct such Riemann domains in the general case.

Question 1. Let H be a smooth Levi-flat hypersurface in ℂ^{n}, n ≥ 3. Let M be a smooth (2n − 2)-manifold contained in H ∩ ∂𝔹^{n} such that its boundary ∂M is the finite union of disjoint connected compact maximally complex CR-manifolds of dimension 2n−3. Let M̃ be a smooth generic globally minimal CR-manifold such that M^{≂} is close (in C^{4}) to M̄ and has the same boundary as M. Is E(∂𝔹^{n}\ M^{≂}) two-sheeted?

Question 2. Can one extend the theorems to the case when 𝔹^{n} is replaced by any smoothly bounded strictly pseudoconvex domain G (not necessarily such that H^{2}(G, ℤ) = 0)?

Question 1 seems to be related to the following question asked by Tomassini [To].

Question 3. Let H be a relatively closed Levi-flat hypersurface contained in 𝔹^{n}(n ≥ 2). Does it have a Stein neighbourhood basis?

Question 4. Let K be as in theorem 1 or be equal to M^{≂}in examples 1 or 2. What is the relation between the polynomial hull K̂ of K and the closure of the set over which the essential envelope of holomorphy of 𝔹^{n} \ K is multi-sheeted?

Note that the latter set is contained in K̂ (see [ČSt]).

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B. Jöricke Department of Mathematics University of Uppsala S-751 06 Uppsala SWEDEN

e-mail: joericke@math.uu.se

N. V. Shcherbina Department of Mathematics University of Göteborg S-412 96 Göteborg SWEDEN

e-mail: nikolay@math.chalmers.se