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SEMI LOCAL LEVI-FLAT EXTENSIONSNIKOLAY SHCHERBINA ∗ AND GIUSEPPE TOMASSINI ∗∗Contents1. Introduction.Let G be a domain in C z × R u ⊂ C 2 z,w (w = u + iv). Let ϕ : bG → R vbe a continuous function and Γ(ϕ) its graph. The extendability of ϕ toa continuous function Φ: G → R v with the Levi-flat (i.e. foliated byholomorphic 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 andstrictly pseudoconvex (i.e. G × R v is a strictly pseudoconvex domainof C 2 z,w). In this paper we study a semi-local version of the extensionproblem, namely, when the function ϕ is prescribed on an open subsetU of bG, where bG is smooth and strictly pseudoconvex. Namely,we define a notion of the hull E(U) ⊂ G of U and prove that everycontinuous function ϕ : U → R v has a Levi-flat extension to E(U).Given an open subset D of C n we denote A(D) the Fréchet algebraO(D) ∩ C 0 (D). If K is a compact subset of D let us denote by ̂K A(D)(or simply by ̂K) the A(D)-hull of K.If D is a strictly pseudoconvex domain in C n and V is an opensubset of bD, we define the set E(V ) to be equal to the union of thehulls ̂K A(D) for all compact subsets K of V . It is easy to see thatif K 1 ⊂ K 2 ⊂ K 3 ⊂ ··· is an exhaustion of V by compact subsets,then E(V )= ⋃ ̂K n . Denote also by E(V ) the CR-hull of V . Since thendomain D is assumed to be strictly pseudoconvex, CR-functions definedDate: November 20, 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 ComplexManifolds”.1

2 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIon V admit holomorphic extension to a one-sided neighbourhood V ofV in the closed domain D and then E(V ) is just the union of V withthe hull of holomorphy of the interior part of V.The first result of this paper describes some properties of the setE(V ).Theorem 1. Let D be a strictly pseudoconvex domain in C n+1z,w , wherez =(z 1 ,z 2 , ..., z n ),w = u + iv, and let V be an open subset of bD. Then1) E(V ) is an open subset of D such that E(V )∩bD = V and E(V )∩Dis pseudoconvex.If D is of the form D = G × R v , where G is a domain in C n z × R u , andV is of the form V = U × R v , where U ⊂bG, then2) E(V ) is invariant under translation in v-direction. In particular,there is an open subset E(U) of G such that E(V )=E(U) × R v .If, moreover, n =1and U is the union of simply-connected subdomainsof bG, then3) E(U × R v )=E(U × R v ).In the case when G is a topological 3-ball the set E(U) enjoies thefollowing property.Theorem 2. Let G ⊂ C z × R u be a domain diffeomorphic to a 3-ballsuch that G × R v ⊂ C 2 z,w is strictly pseudoconvex. Let U be an opensubset of bG and let {U α } be the connected components of U. Then thesets E(U α ) are connected and disjoint and, moreover, E(U) = ⋃ E(U α ).αNote that the statement of Theorem 2 is in general false if G is notdiffeomorphic 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 constructionof the hull of holomorphy of a rigid domain was given in theclassical 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 boundaryof strictly pseudoconvex domain G × R v , it is possible to provide aconcrete geometric description of its CR-hull. Such description is givenin Section 4 of this paper and is used later in Example 2 presented inSection 7.

SEMI LOCAL LEVI-FLAT EXTENSIONS 3Let, as above, G ⊂ C z × R u be a domain such that G × R v ⊂ C 2 z,wis strictly pseudoconvex, U an open subset of bG and ϕ : U → R v acontinuous function. Consider the setsV + ϕ= {(z,w)∈U × R v : ϕ(z,u)

4 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIdifference that one of the functions h − of −h + can be identically equalto −∞. Following the notations in [CS] we denote bybG ± = {(z,u) :z = π(ζ), u= h ± (ζ), ζ∈G}respectively the “upper” and the “lower” part of bG (if h − ≡−∞orh + ≡ +∞, 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 strictlypseudoconvex. Let U be an open subset of bG such that each connectedcomponent 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 acompact subset of U α ′ . Then there is a function ϕ ∈ C(U) which doesnot admit a continuous Levi-flat extension to any domain Ω ⊂ G suchthat E(U) Ω.Remark. It is easy to see that in the case when the domain G isdiffeomorphic to a 3-ball it is enough to assume that each connectedcomponent U α of U is contained either in bG − or bG + .Remark. The main part of our results holds true for domains G withmuch weaker regularity assumptions as in [CS] and [ST]. We presentthem for strictly pseudoconvex case to avoid additional technical difficulties.Acknowledgements. Part of this work was done while the first author wasa visitor at Scuola Normale Superiore (Pisa)and at the Max Planck Instituteof Mathematics (Bonn). It is his pleasure to thank both institutions for theirhospitality and excellent working conditions.2. Preliminaries.Lemma 1. Let D be an unbounded strictly pseudoconvex domain inC n . Let K be a compact subset of bD. Then there exists a boundedstrictly pseudoconvex subdomain D ′ of D such that K ⊂ bD ′ and theA(D)-hull ̂KA(D) is equal to the A(D ′ )-hull ̂KA(D ′ ) of K. In particular,in the case n =2, this implies that the set D ̂K A(D) is pseudoconvex.

SEMI LOCAL LEVI-FLAT EXTENSIONS 5Proof. Let B R = {z ∈ C n : |z| 0 there exists a function g holomorphic in a neighbourhood W ofD ′ such that ||g − f|| ′ D 0 such that for any vector

6 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIe of length 1 one has (K n + δe) ∩ D⊂V n+1 and ( ̂K n + δe) ∩ bD⊂K n+1for all δ, 0≤ δ

SEMI LOCAL LEVI-FLAT EXTENSIONS 7smooth simply-connected domains D p n ⋐U such that E n ⊂∪ p D p n. Considerpiece-wise smooth 2-spheresS p n =((bD p n) × [−n − 1,n+ 1]) ∪ (D p n ×{−n − 1}) ∪ (D p n ×{n +1})contained in bG, their smooth generic perturbations ˜S n p ⊂bG × R v andgiven by Kruzhilin [Kr] 3-balls Bn p with the boundary ˜S n p foliated byholomorphic discs. Since bBn p = ˜S n p ⊂ bG × R v K n , we concludefor each p from pseudoconvexity of the domain G × R v ̂K n and from“Kontinuitätsatz” that Bn∩ p ̂K n = ∅ and that there is a neighbourhoodVn p of Bn p in G × R v such that Vn p ⊂ E(U × R v ). Denote by Wnpthe bounded component of the set G × R v Bn.p Then, by strictpseudoconvexity of G × R v , there is a neighbourhood Wn p of (bG ×R v ) ∩ W p n in G × R v which is contained in E(U × R v ). Since Vn p ⊂E(U × R v ) and Wn p ⊂ E(U × R v ), we conclude from Hartogs theoremon removability of compact singularities that Wn p ⊂ E(U × R v ) for allp. Then the inclusion ̂K n ⊂ ⋃ p Wp n implies that ̂K n ⊂ E(U × R v ) andhence E(U × R v )= ⋃ ̂K n n ⊂ E(U × R v ). The proof of Theorem 1 isnow completed. ✷Proof of Theorem 2. We consider first the special case when thedomain G is bounded. Then the set E(U) cannot have connected componentswith part of the boundary in two different components U α1and U α2 . Indeed, let F be the connected component of bU α1 such thatU α1 and U α2 are contained in different connected components of theset bG F . Since F is connected, and since bG is diffeomorphic toa 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 sucha way that U α1 ⊂ V 1 . Let D 1 ⊂ D 2 ⊂··· be an exhaustion of theset bG \ F by open subsets such that for each k =1, 2, ... one hasD k ⋐ D k+1 , the set D k is the union of finitely many simply-connecteddomains with smooth boundary and for each β the set Dβ k = Dk ∩ V βconsists of at most one connected component. Observe that accordingto our enumeration for the domains {Dβ k} one has U α 1⊂ +∞ ⋃D1.kLet E 1 ⊂E 2 ⊂··· be an exhaustion of U by compact subsets such thatE n ⋐ o En+1 and E n ∩U α is connected for every n∈N and every α. Let ̂K nbe the A(G × R v )-hull of K n = E n ×[−n, n]. Since ⋃ D k = bGF ⊃U,kit 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 smoothk=1

8NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINI2-sphereS k(n)β=((bD k(n)β)×[−n−1,n+1])∪(D k(n)β×{−n−1})∪(D k(n)β×{n+1})contained in bG. Smoothing the edges of S k(n)βand perturbing it a littlek(n)bit we get a smooth generic 2-sphere ˜Sβ⊂ bG. It follows then fromKru˘zilin [Kr] (see also Bedford-Klingenberg [BK]), that there is a 3-ballB k(n)k(n)βwith the boundary ˜Sβfoliated by holomorphic discs. Since,by our construction, the domains D k(n)βare disjoint, the 3-balls B k(n)βwill also be disjoint. Since each ball B k(n)βis foliated by holomorphicdiscs with the boundary bS k(n)β⊂bG K n , we conclude from Lemma 1and“Kontinuitätsatz” that B k(n)β∩ ̂K n = ∅.For each β denote by W k(n)βthe bounded component of the set G ×R v B k(n)β+∞⋃n=1. Then we have ̂K n ⊂ ⋃ β⋃W k(n)1 and W 2 = +∞ ⋃n=1 β≠1W k(n)β. Consider the sets W 1 =W k(n)β. Since for each n the sets W k(n)βarerelatively open in G × R v and disjoint, and since for each β the familyof sets W k(n)βis increasing with respect to n, it follows that the setsW 1 and W 2 are disjoint and relatively open in G × R v . The inclusion̂K n ⊂ ⋃ W k(n)βfor each n implies that E(U × R v )= ⋃ ̂K n ⊂W 1 ∪W 2 .β nSince U α1 × R v ⊂W 1 and U α2 × R v ⊂(U U α1 ) × R v ⊂W 2 , we concludethat the set E(U) cannot have connected components with parts of theboundary in two different connected components U α1 and U α2 .In order to prove that E(U) = ⋃ E(U α ) it is enough to show that forαeach connected component U α1 of U one has E(U) ∩W 1 = E(U α1 ) (seethe definition of W 1 above). Since E(U)×R v = ⋃ ̂K n and E(U α1 )×R v =n⋃n̂K Uα 1n (here K Uα 1n= K n ∩ (U α1 × R v )), it is sufficient to prove that foreach n we have that ̂K n ∩W k(n)1 = ̂K Uα 1n . It follows from Lemma 1 thatfor each n the set W k(n)1 ( ̂K n ∩W k(n)1 ) is pseudoconvex and, therefore,̂K n ∩W k(n)1 ⊂ ̂K Uα 1n . The opposite inclusion ̂K Uα 1n ⊂ ̂K n ∩W k(n)1 isobvious, since ̂K Uα 1n ⊂ ̂K n . Hence ̂K n ∩W k(n)1 = ̂K Uα 1n for each n andthenE(U) ∩W 1 = ⋃ n( ̂K n ∩W k(n)1 )= ⋃ n̂K Uα 1n = E(U α1 ).

SEMI LOCAL LEVI-FLAT EXTENSIONS 9The connectedness of the sets E(U α ) follows from the fact that E(U α )×R v = ⋃ ̂K UαUαn and the connectedness of each set ̂K n which is duento Shilov Idempotent Theorem (see, for example [G], p. 88). Thiscompletes 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 isan exhaustion of G by bounded domains G n such that G n × R v arepiece-wise strictly pseudoconvex and G ∩ B n ⊂ G n , where B n is theball in C z × R u of radius n with centre at the origin. Moreover, sincethe domain G is assumed to be diffeomorphic to a 3-ball, the coveringmodel G of G is simply-connected, hence, we can use in the mentionedabove construction of G n an exhaustion of G by simply-connected subdomainsto 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 bybounded domains ˜G n diffeomorphic to a 3-ball such that ˜G n × R v arestrictly pseudoconvex and G∩B n ⊂ ˜G n . Denote by U n the interior partof the set U ∩ b ˜G 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 withrespect to ˜G n . It is also easy to see that E(U) = +∞ ⋃E(U n ). Therefore,applying the argument of the first part of the proof to ˜G n and U n andpassing to the limit as n →∞we obtain the required properties ofE(U) in the general case. The proof of Theorem 2 is now completed.✷n=1Remark. In the proof of Theorem 2 the use of Bedford-KlingenbergTheorem 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 inC z × R u such that G × R v is strictly pseudoconvex. Since the CR-hullE(U × R v )ofU × R v is equal to the union of CR-hulls of its connectedcomponents, and since for any exhaustion U 1 ⋐ U 2 ⋐ U 3 ⋐ ··· of U onehas E(U × R v )=∪ ∞ n=1E(U n × R v ), it is enough to give a descriptionof the CR-hull of U × R v for U being a smooth bounded domain in bGwhich is in general position with respect to the canonical projection

10 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIp : G → G. The last assumption means that all the selfintersectionsand intersections with bG of the curve p(bU) are transversal. In the restof this section we will always assume that U ⊂bG is a smooth boundeddomain in general position. In the case of an arbitrary domain U acharacterization similar to the given below in Proposition 1 can beeasily obtained by passing to the limit.Let U − = U ∩ bG − and U + = U ∩ bG + , where bG − and bG + arethe ”lower” and the ”upper” part of bG, respectively, defined in theIntroduction. Denote by Σ the union of all connected components ofthe set p(U − ) ∩ p(U + ) whose closure has a nonempty intersection withbG. Further, denote U − = p(U − ) Σ and U + = p(U + ) Σ. Considerthe function f − defined on b U − which equals h − on the set b U − bΣand h + on the set b U − ∩ bΣ. Let F − be the harmonic extension off − to U − . Similarly, consider the function f + defined on b U + whichequals 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 + . DenoteandΣ = {(z,u) :z = π(ζ), h − (ζ) ≤ u ≤ h + (ζ), ζ∈ Σ},U − = {(z,u) :z = π(ζ), h − (ζ) ≤ u

SEMI LOCAL LEVI-FLAT EXTENSIONS 11projection p : G → G. Then for the CR-hull E(U × R v ) of the setU × R v one has E(U × R v )=U × R v .Proof. We show first that the CR-hull E(U × R v )ofU × R v is singlesheeted,i.e. it is a subdomain of 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 vsuch that W n ⊂ bG n . It follows then from [St] that E(W n ) is equal tothe complement in 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 theincreasing sequence of hulls E(W n ), we conclude that E(U ×R v ) is alsosingle-sheeted.Now we observe that the domain U × R v is pseudoconvex. Indeed,pseudoconvexity on the part (U ∩ bG) × R v of its boundary followsfrom strict pseudoconvexity of G × R v . For proving pseudoconvexityof U × R v inside G × R v we denote by I the set of those points ofselfintersection of p(bU) which are contained in Σ. Further, we denoteby I the set{(z,u) ∈ G : z = p(ζ),h − (ζ)

12 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIdiscs. More precisely, for each point Q ∈ Σ we construct a family ofholomorphic discs {D t },0≤ t ≤ 1, such that D 0 ⊂V×R v , bD t ⊂V×R vfor all t ∈ [0, 1] and D t ∩ ({Q}×R v ) ≠ ∅ for some t. Since the surfaceU ×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 ofholomorphic discs. Namely, for each point Q ∈ U − ∪ U + we constructholomorphic 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 vas it is stated in proposition.Step 1. Let Q be a point of the set Σ. To simplify the proof weassume that the set Σ is connected. In the general case one can applythe argument below to each of the connected components of Σ. Considera point ζ 0 contained in the interior of one of the arcs constitutingthe set b G∩Σ. Since the set Σ = p(Σ) is connected, there is a smoothcurve γ(t) ⊂ Σ, 0 ≤ t ≤ 1, connecting the points ζ 0 and p(Q) ∈ Σ andtransversal to bG at the point ζ 0 . For each T ≤ 1 denote by γ Tthepart of γ corresponding to t ∈ [0,T]. Let W be a neighbourhood of ζ 0in G so small that p −1 (W)⊂V. Choose δ>0 small enough so that forthe δ-neighbourhood V δ (γ) ofγ in G one has V δ (γ) ∩ bΣ⊂W. ConsiderT 0 < 1 such that V δ (γ T)⊂W for all T ∈ [0,T 0 ]. For δ 1 ∈ (0,δ) and foreach T ∈ [T 0 , 1] consider the domain A T = V δ (γ T) V δ1 (γ T) ⊂ G. Theboundaries of the domains A T are smooth everywhere except for somepoints of the set bΣ ∩W where they have transversal intersection withbG. 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) onbA T to be equal toh − (ζ) onbV δ (γ T) ∩G, to be equal to h + (ζ) onbV δ1 (γ T) ∩G and finallyto be equal to h − (ζ) =h + (ζ) onbA T ∩ bG. Then {φ T } is a familyof functions of Hölder class C 1/2 (bA T ) depending continuously on T(each of these functions will behave like ± √ dist(ζ,bG) asζ → bG).Let F T be the harmonic extension of φ T to A T . Since each domain A Tis constructed to be simply-connected, for each T ∈ [T 0 , 1] there existsa holomorphic function G T defined on A T and continuous A T such thatRe G T = F T (these functions will actually be also in some Hölder classon ĀT ). One can obviously choose the functions G T to be continuouslydependent of T . Consider the family of holomorphic discsD T = {(z,w) :z = π(ζ),w = G T (ζ), ζ∈ ĀT }, T 0 ≤ T ≤ 1.Then, by our construction, D T depend continuously on T , for each Tone has bD T ⊂ U × R v and, since A T0 ⊂W, we conclude that D T0 ⊂V × R v . Moreover, the family of domains A T and the functions F T

SEMI LOCAL LEVI-FLAT EXTENSIONS 13were constructed in such a way that the graph of some function F T inour family have to contain the point Q and, hence, the correspondingholomorphic disc D T will contain a point of {Q}×R v . It follows thenfrom “Kontinuitätssatz” that this point of {Q} ×R v is contained inE(U × R v ). Since the surface U × R v is rigid, the whole line {Q}×R vis contained in E(U × R v ), and since Q is an arbitrary point of Σ, wefinally conclude that Σ × R v ⊂E(U × R v ).Step 2. Here we need to prove that U − ∪ U + ⊂ E(U × R v ). Sincethe argument for both sets U − and U + is the same, we will show thatU − ⊂ 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 connectedcomponent of U − separately. In particular, our assumption impliesthat U − is a (connected) domain in 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 theu-direction) part of b U − . Remind, that by definition of the functionf − = F − | b U − it is continuous everywhere except maybe for finitelymany points contained in b U − ∩ b U + .Now we consider a new domain Ũ − obtained by shrinking the domainU − on the part b U − bΣ of its boundary and enlarging it on theresting part b U − ∩ bΣ of the boundary so that the domain Ũ − willhave a smooth boundary. Further, we approximate the function f − bya smooth function ˜f − defined on b Ũ − . More precisely, let Ũ − be asmoothly bounded domain over G with projection ˜π : Ũ − →Gand let˜f − be a smooth function on b Ũ − with harmonic extension ˜F − to Ũ −such that ˜π(Ũ − )⊂U − ∪ Σ,Γ( ˜f − ) def= {(z,u) :z = π(˜π(ξ)),u= ˜f − (ξ),ξ ∈ b Ũ − }⊂U − ∪ Σand ˜F − (ξ Q ) >F − (ζ Q ) − ε/2, where ξ Q =˜π −1 (ζ Q ). The last inequalityimplies that u Q < ˜F − (ξ Q ) − ε/2. The problem now is that the domainŨ − might be multi-connected, while for being able to use “Kontinuitätssatz”we need to have a simply-connected domain. That is whywe need to modify the domain Ũ − and the function ˜f − further. To dothis we use the argument of Lemma 5.1 in [ST]. For the convenience ofreading we remind the construction used there.Let Γ 1 , Γ 2 ,... ,Γ m be the connected components of b Ũ − . Considerthe universal covering ˜π :∆→ Ũ − of Ũ − by the unit disc ∆. Letγ 1 ,γ 2 ,... ,γ m−1 be a disjoint family of smooth curves in Ũ − such thateach curve γ i , i =1,... ,m − 1, connects the corresponding curves

14 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIΓ 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 ofb U − is contained in Σ, we can choose the curves γ i to be contained in˜π −1 (U − ). Then the domain Ũ − \ ⋃ m−1i=1 γ i is simply-connected. Hence,the set ˜π −1 (Ũ − \ ⋃ m−1i=1 γ i) is the disjoint union of a countable familyof simply-connected subdomains {F σ } of ∆. For each σ denote byFσ ∗ the union of the set F σ ∩ ∆ with all the domains F ′ σsuch thatF σ ∩ F σ′ ≠ ∅.Choose now one of the domains {F σ } and denote it by F 0 . Denoteby F 1 the corresponding domain F0 ∗ . Define inductively for each n =2, 3,... a domain F n as the union of all the domains Fσ ∗ 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 =∆.The following lemma was proved in [ST] (Lemma 5.1).Lemma. There exists a positive number θ ˜F − (ξ Q ) − ε/2 ifn is big enough. In particular,u Q < ˜Fn − (η Q ). Note, that by the construction of F n and ˜Fn − one has˜π(˜π(F n ))⊂ U − ∪ Σ andΓ( ˜Fn − ) def= {(z,u) :z = π(˜π(˜π(η))),u= ˜fn − (η),η ∈ bF n }⊂U − ∪ Σ.Now we are in position to construct the family of holomorphic discsrequired by “Kontinuitätssatz”. Define for t ∈ [1, 2] the function φ ′ t onbF n as φ ′ t (η) =(1− t) ˜fn − +(2− t)h − (˜π(˜π(η))) and for each t considerthe harmonic extension ˜Ft of the function φ ′ t to the domain F n . SinceF 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 andcontinuous on F n which depend continuously on the parameter t and

SEMI LOCAL LEVI-FLAT EXTENSIONS 15such that Re G ′ t = ˜Ft . Consider the family of holomorphic discsD ′ t = {(z,w) :z = π(˜π(˜π(η))),w = G ′ t (η),η ∈ F n }, 1 ≤ t ≤ 2.Then, by our construction of F n and φ ′ t , for each t ∈ [1, 2] one hasbD ′ t ⊂ (U − ∪ Σ) × R v and, since ˜F1 (η) =h − (˜π(˜π(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 smoothlybounded simply-connected subdomains A ′ t ,0≤ t ≤ 1, of F n which dependscontinuously on t and such that A ′ 1 = F n and A ′ 0 is a point inF n . For each t ∈ [0, 1] let ˜Ft be the harmonic extension of the functionφ ′ t (η) =h − (˜π(˜π(η))) from bA ′ t to A ′ t . Consider a one-parameterfamily of holomorphic functions G ′ t defined on the respective domains′′A t which depend continuosly on t and such that Re G t = ˜Ft . Thenfor the family of holomorphic discsD ′ t = {(z,w) :z = π(˜π(˜π(η))),w = G ′ t(η),η ∈ Ān},′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 ofdiscs D ′ t ,0≤ t ≤ 2, then it will depend continuously on t, D 0 will beapointinU × R v and for each t one will have bD ′ t ⊂ (U − ∪ Σ) × R v .Observe now that by our construction one has⋃π z,u (D ′ t )={(z,u): π(˜π(˜π(η))),h − (˜π(˜π(η))) ≤ u ≤ ˜Fn − (η),η ∈F n },0≤t≤2where π z,u : C 2 z,w → C z × R u is the projection. Since u Q < ˜Fn − (η Q ), itfollows that there is a disc D ′ t0 in our family such that Q ∈ π z,u (D ′ t0 ).This means that there is a point of {Q} ×R v contained in D ′ t0 . Itfollows then from “Kontinuitätssatz” that this point is contained inthe 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 nowthat the line {Q} ×R v is contained in E(U × R v ) and then, by thechoice of Q, the whole set U − ∪ U + is contained in E(U × R v ). Thiscomplete 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 G×R vfollows from the part 1 of Theorem 1 and the fact that these sets can

16 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIbe represented in the form (1) and (2) follows from the same argumentas in the part 2 of Theorem 1. The property Φ ± | U = ϕ follows directlyfrom the definition of E(V ± ϕ ). ✷Proof of Theorem 4. We start with the proof of continuity of thefunctions Φ ± at the points of the set U. Since the domain G × R v isstrictly pseudoconvex, there are one-sided neighbourhoods V + and V −of the sets V ϕ + and Vϕ − in G × R v , respectively, filled by holomorphicdiscs with the boundary on the respective set V ϕ+ and Vϕ − . Since V + ⊂E(V ϕ + ) and V − ⊂E(Vϕ − ), 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 ± arecontinuous at the points of U.To prove continuity of the functions Φ ± in the interior points of E(U)we need the followingLemma 2. There is an increasing sequence of open relatively compactsubsets {E n } of E(U) such that1) E n × R v is piece-wise strictly pseudoconvex for every n ∈ Nand2) For each compact subset K of E(U) one has K ⊂ E n for all n bigenough.Proof. Let, as above, E 1 ⊂ E 2 ⊂ E 3 ⊂··· be an exhaustion of U bycompact subsets and let K n = E × [−n, n]. Since E(U) = ⋃ π z,u ( ̂K n ),nwhere π z,u : C 2 z,w → C z × R u is the projection, it is enough to constructfor each n ∈ N an open subset E n of E(U) satisfying property 1 andsuch that π z,u ( ̂K n ) ⊂ E n . For n being fixed, we can assume in theconstruction of E n that the domain G is bounded. If not, we use thesame argument as in Lemma 1. Namely, for r large enough considerthe 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˜G r such that ˜G r × R v is strictly pseudoconvex. Finally, we can replacethe domain G by ˜G r and the set U by the interior part of the setU ∩ b ˜G r .Let ϱ be the defining function of the domain G, i.e. G = {(z,u) :ϱ(z,u) < 0} and grad ϱ ≠0onbG. For each δ>0 consider the domainG δ = {(z,u) :ϱ(z,u)

SEMI LOCAL LEVI-FLAT EXTENSIONS 17where ε n = dist(E n ,bU) and E n is the corresponding compact from theexhaustion of U. Let E(Un) δ be the set defined in the same way as theset E(U) with G and U replaced by G δ and Un, δ respectively. Since forall δ small enough the domains G δ ×R v are strictly pseudoconvex, thereare uniformly big one-sided neighbourhoods Vn δ of the sets Un δ × R v inthe respective domains G δ × R v filled by holomorphic discs with theboundary on the corresponding set Un δ ×R v . Then, for some δ 0 > 0 onehas K n ⊂ V δ 0n ⊂E(U δ 0n ) × R v . Therefore, by the definition of E(U δ 0one also has that ̂K n ⊂E(U δ 0n ) × R v , i.e. π z,u ( ̂K n )⊂E(U δ 0n ).It also follows from strict pseudoconvexity of the domains G δ × R vthat there are uniformly big one-sided neighbourhoods Ṽ δ n of the sets(bG δ U δ n)×R v in the respective domains G δ ×R v filled by holomorphicdiscs with the boundary on the corresponding set (bG δ U δ n) × R v .Since, by definition, the set E(Un) δ × R v is the union of the hulls ofcompacts exhausting the set Un δ ×R v , and since, by Lemma 1, the discsfilling Ṽ δ n cannot intersect these hulls, it follows that bG ∩E(Un) δ ⊂ Uand, therefore, E(Un)∩G⊂E(U) δ for all δ small enough. We can assumethat this inclusion holds for δ 0 chosen above (otherwise we choose thesmallest of these two δ’s). In view of Theorem 1, the interior part ofthe set E(U δ 0n ) × R v is pseudoconvex, hence the functionh(z,u) =− log dist((z,u),bE(U δ 0n ) × R v )is plurisubharmonic in the interior part of the set E(U δ 0n )×R v . Smoothingthis function and adding the function θ(|z| 2 +u 2 ), θ>0 being smallenough, we get a strictly plurisubharmonic function ˜h(z,u). Chooseη>0 sufficiently big such that the domain E η (U δ 0n )={(z,u) :˜h(z,u) 0 definethe 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 = ton B n,|t| and ϕ n,t1 ≤ ϕ n,t2 if t 1 ≤ t 2 . Then, in view of the constructionn ),

18NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIof E n , we can apply Theorem 2 of [CS] to the functions ϕ n,t and obtainfor each t a function Φ n,t ∈ C(E n ) such that Φ n,t = ϕ n,t on bE n and thegraph Γ(Φ n,t ) Γ(ϕ n,t ) is (locally) foliated by 1-dimensional complexsubmanifolds. Define functions ˜Φ ± on E(U) as˜Φ + (P ) =limn→+∞lim Φ n,t(P ) and ˜Φ − (P ) =t→+∞limn→+∞lim Φ n,t(P ).t→−∞Remark. We will prove later (see Lemma 5) that the functions ˜Φ ±coincide with Φ ± , respectively, and then we only use the notation Φ ± .In the next lemma we prove the uniform boundedness of the functionsΦ n,t on compact subsets of E(U).Lemma 3. For each compact subset K of E(U) there exist a constantC>0 and a number N such that |Φ n,t (P )| ≤C for all P ∈ K, t ∈ Rand 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, suchthat for every n ∈ N each set K n± is a subset of the interior part ofthe respective set K n+1. ± Since, by Theorem 3, π z,u (E(V ϕ ± )) = E(U), itfollows that there is a number L such that K ⊂ π z,u ( ̂K ± L). Consider anumber N such that π z,u ( ̂K ± L ) ⊂E N. Then for each n ≥ N and t ∈ Rthe graph Γ(Φ n,t ) of the function Φ n,t does not intersect the sets ̂K ± L .Since the argument is the same for both sets ̂K ± L, we prove this propertyfor the set ̂K + L .Indeed, for n ≥ N and t ∈ R being fixed, it follows from continuityof Φ n,t on E n and compactness of ̂K+L that Γ(Φ n,t − C) ∩ ̂K + L = ∅for C>0 big enough. Since, by definition of ϕ n,t and the choice ofE N ,Γ(ϕ n,t − s) ∩ ̂K + L= ∅ for s ≥ 0, and since, by Lemma 1, thedomain G × R v ̂K + Lis pseudoconvex, we conclude from Levi-flatnessof Γ(Φ n,t − s), 0 ≤ s ≤ C, that Γ(Φ n,t ) ∩ ̂K + L = ∅.To complete the proof of Lemma 3 we observe that the propertiesΓ(Φ n,t ) ∩ ̂K ± L = ∅ and K ⊂ π z,u( ̂K ± L ) imply that |Φ n,t(P )| ≤C for allP ∈ K, t ∈ R and n ≥ N, where C = max{|v| :(z,u + iv) ∈K ± L }. ✷

SEMI LOCAL LEVI-FLAT EXTENSIONS 19To show that the graphs Γ(˜Φ ± ) Γ(ϕ) are foliated by holomorphiccurves we follow the argument of section 6 in [ST]. Namely, fromLemma 3 we conclude, using Lemmas 3.2-3.5 in [Sh], that for eachpoint P =(z 0 ,u 0 ) ∈E(U) ∩ G there exists R ∗ > 0 (which depends onlyon dist (P, bE(U)) and the constant C given by Lemma 3 for compactK being a fixed neighbourhood of P ) such that for each t ∈ R and alln big enough there is a holomorphic functionfn,t(z) P =u P n,t(z)+ivn,t(z) P :∆ R ∗(z 0 ) → C wwith the properties u P n,t(z 0 )=u 0 and Γ(fn,t)⊂Γ(Φ P n,t ), where ∆ R ∗(z 0 )={|z − z 0 | 0. It follows then from ourassumption and from continuity of ˜Φ ± near U, that π z,u (L) ∩ V ⋐ G.Since, by Lemma 3, ˜Φ− is bounded over compact subsets of E(U),we have from the construction of holomorphic leaves in Γ(˜Φ − ) thatthe closure L of the leaf L in V × R v is compact and that the setL∩(V × R v ) is foliated by holomorphic curves. Therefore, the domainV × R v L is pseudoconvex near L. Then, considering the functionϱ(P )=− log dist(P, L)+ε(|z| 2 + |w| 2 ) with small enough ε>0 andsmoothing it, if necessary, we construct (as a superlevel set of the

20 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIconstructed function) a strictly pseudoconcave neighbourhood V of Lin V ′ × R v (we mean here the strict pseudoconcavity of V near thepart bV ∩(V ′ × R v ) of its boundary) such that V ⋐ G × R v , whereV ′ is a slightly smaller than V neighbourhood of π z,u ( ̂K n )inE(U).Since the set V is compact, there is a big enough constant C such that(V +Ce v )∩ ̂K n = ∅. Then the property π z,u (L)∩π z,u ( ̂K n ) ≠ ∅ impliesthat for some t ≥ 0 the set V +(C −t)e v will touch for the first momentthe set ̂K n . This contradicts pseudoconvexity of G × R v \ ̂K n given byLemma 1. ✷Continuity of the functions ˜Φ ± in the interior points of E(U) followsnow directly from the argument of Lemma 7.2 in [ST] with Lemma7.1 replaced by just proved Lemma 4. Hence, we have proved thatthe functions ˜Φ ± define continuous Levi-flat extensions of the givenfunction ϕ ∈ C(U) to the domain E(U).Lemma 5. The functions Φ ± defining the sets W ± coincide with therespective functions ˜Φ ± .Proof. Since the argument is the same for both functions, we provethe statement for the function Φ + . First, we will show that Φ + ≤ ˜Φ +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 ofV ϕ + , ϕ m,t and Theorem 3 that for m, ε and t (big enough) being fixedone has Γ(ϕ m,t + ε) ⊂ ̂K n + if n is sufficiently big. Therefore, by themaximum principle, we conclude from the Levi-flatness of Γ(Φ m,t + ε)that Γ(Φ m,t + ε) ⊂ ̂K n + . This implies, by the definition of ˜Φ + andE(V ϕ + ), that Γ(˜Φ + + ε) ⊂ E(V ϕ + ). Hence, Φ + ≤ ˜Φ + + ε for each ε>0and, therefore, Φ + ≤ ˜Φ + .To prove the other inequality ˜Φ + ≤ Φ + we argue by contradictionand assume that Φ + (P 0 ) < ˜Φ + (P 0 ) for some point P 0 ∈ E(U). Itfollows then from the definition of Φ + and E(V ϕ + ) that there exist anumber n 0 ∈ N and a point (P 0 ,v 0 ) ∈ ̂K n + 0such that v 0 < ˜Φ + (P 0 ).Therefore, by the definition of ˜Φ + , there are m 0 ∈ N and t 0 ∈ R suchthat v 0 < Φ m0 ,t 0(P 0 ). We can also assume that m 0 is chosen so big thatthe domain(E m0 (where the function Φ m0 ,t 0is defined) has the propertydist bE m0 ∩ G, π z,u ( ̂K)n + 0) > 0. Since Γ(Φ m0 ,t 0− C) does not intersect

SEMI LOCAL LEVI-FLAT EXTENSIONS 21̂K n + 0for C big enough, and since, by the choice of m 0 ,Γ(ϕ m0 ,t 0− s)does not intersect ̂K n + 0for all s ≥ 0, we conclude from pseudoconvexityof the domain G × R v ̂K n + 0(see Lemma 1) and from Levi-flatnessof Γ(Φ m0 ,t 0− s), 0 ≤ s ≤ C, that Γ(Φ m0 ,t 0− s) ∩ ̂K n + 0= ∅ for all0 ≤ s ≤ C. This contradicts the inequality v 0 < Φ m0 ,t 0(P 0 ) and provesLemma 5. ✷The last statement of Theorem 4, saying that for each functionΦ ∈ C(E(U)) such that Φ| U = ϕ and Γ(Φ) Γ(ϕ) is Levi-flat onehas Φ − (z,u) ≤ Φ(z,u) ≤ Φ + (z,u) for all (z,u) ( ∈E(U), follows frompseudoconvexity of the domains G × R v ̂K− n ∪ ̂K)n + by the sameargument as in the proof of Lemma 3. The proof of Theorem 4 is nowcompleted. ✷Remark. Denote U c = bG \ U and let E ∗ (U) be the union of allconnected components of the set E(U ∪ U c ) which have some pointsof U on the boundary. Consider a function ψ ∈ C(U c ). ApplyingTheorem 4 to the set U ∪ U c instead of U and to the function on U ∪ U cwhich is equal to ϕ on U and to ψ on U c , we obtain Levi-flat extensionsof ϕ to the set E ∗ (U) which is in general bigger than E(U) (see Example1 in Section 7). If the domain G is diffeomorphic to a 3-ball, then, inview of Theorem 2, E ∗ (U) =E(U) and these Levi-flat extensions ofϕ 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 belowto all the connected components of U. Since both cases U ⊂ bG − andU ⊂bG + can be treated in the same way, in what follows we assume thatU ⊂ bG − . Consider an exhaustion U 1 ⋐ U 2 ⋐ U 3 ⋐ ··· of U by domainswith smooth boundary and for each n =1, 2,... let U n = p(U n ), wherep : G → G is the canonical projection. Further, consider U ′ = p(U ′ ),where U ′ ⊂bG − is the simply-connected domain given by assumption ofthe theorem. Denote by A n the set consisting of all compact connected

22 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIcomponents of U ′ ′ U n . Then the sets U n = U n ∪A n are simplyconnectedsubdomains of U ′ ′ ′ ′such that U 1 ⋐ U 2 ⋐ U 3 ⋐ ···⊂U ′and U ′ = ∪ +∞n=1U ′ ′n . For each n =1, 2,... define the function H n onthe domain U ′ to be equal to h − on the set U ′ A n and to be theharmonic 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 theset 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 subharmonicon U ′ ′ ′ ′ ′′ ′′ ′′and ··· ≤ H 3 ≤ H 2 ≤ H 1 ≤ H 1 ≤ H 2 ≤ H 3 ≤···. LetH ′ (ζ) = lim H n ′ (ζ) and H ′′ (ζ) = lim H n ′′ (ζ), ζ ∈U ′ . Define then→+∞ n→+∞subdomain U of G byU = {(z,u) : z = π(ζ),H ′ (ζ)

SEMI LOCAL LEVI-FLAT EXTENSIONS 23For each n =1, 2,... choose t n ∈ R in such a way that all discsD α n,t nare contained in {v ≤−n}. Further, for (z,u) ∈ bU n defineϕ(z,u) to be equal to ImG α n(ζ)+t n , where ζ is the point of bU n suchthat (z,u) =(π(ζ),h − (ζ)). Since the sequence of sets {bU n } has noaccumulation point in U, we can extend this function to a continuousfunction ϕ defined on the whole of U. We claim that U is the maximaldomain for the Levi-flat extension of the defined function ϕ.Assume, to get a contradiction, that there is a domain Ω⊂G, U Ω,such that our function ϕ has a continuous Levi-flat extension Φto Ω.Consider a point Q ∈ (bUbG)∩Ω. Then, by definition of U, for eachn =1, 2,... we can choose α n in such a way that Q will be containedin the set of accumulation points for the sequence of graphsΓ(F αnn)={(z,u) : z = π(ζ),u= Fnαn(ζ),ζ ∈ B αnn }of functions Fnαn . By our assumption, the functions Φis continuouson the set Γ(Fnαn ) ⊂ Ω. Hence, for c>0 big enough the disc Dn,c αn willnot intersect the graph Γ(Φ) of Φ. Since, by our assumption, Γ(Φ) isLevi-flat, the complement to Γ(Φ) in C 2 z,w is locally pseudoconvex atthe interior points of Γ(Φ). Applying “Kontinuitätssatz” to the familyof discs Dn,t αn, t n ≤ t ≤ c, we conclude that Dn,t αn∩ Γ(Φ) = ∅ for allt>t n . This means that the disc Dn,t αnnis situated above Γ(Φ) in thev-direction. Then the definition of ϕ on bU n and the choice of t n inthis definition imply that Φ ≤−n on Γ(Fnαn ). Now the fact that Q iscontained in the accumulation set of the sequence {Γ(Fnαn )} contradictsto our assumptions that Q ∈ Ω and Φ ∈ C(Ω). ✷In view of Theorems 4 and 5 and the argument of Theorem 2 appliedto the discs D α n,t instead of the Bedford-Klingenberg’s discs, we havethe followingCorollary. Under the assumptions of Theorem 5 the sets E(U α ) aredisjoint and E(U) = ⋃ E(U α ).αRemark. Surprisingly, the domain E(U) will not be maximal for Leviflatextensions from U if we relax our assumption that every connectedcomponent of U is contained either in bG − or in bG + . Namely, thereis a smoothly bounded simply-connected domain U in the boundary ofthe unit ball B of C z × R u such that every function ϕ ∈ C(U) admitsa continuous Levi-flat extension to a subdomain of B which is bigger

24 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIthan E(U). The authors are planning to present an example of suchset 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 theinequality (|z| −2) 2 + u 2 < 1. By a direct computation of the Leviformone can easily see that the domain G × R v ⊂ C 2 z,w is strictlypseudoconvex (see also Theorem 1 in [CS]). Consider the open subsetU = {(z,u) ∈ bG : |z| < 2} of bG. Then U c = bG \ U = {(z,u) ∈bG : |z| > 2}. The sets U and U c are obviously disjoint, but the setE(U ∪ U c )={(z,u)∈G : |u| < 1} is connected (the set E(U ∪ U c ) × R vis foliated by annuli A C = {(z,w) ∈ G × R v : u =ReC, v =ImC}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 followthat there is a uniquely determined 3-dimensional ball B 3 embeddedinto D and foliated by holomorphic discs such that bB 3 = S?Note, that by [BK] and [Kr] there are 2-spheres S ′ ⊂ bD arbitrarilyclose to S which satisfy all the requirements of the problem. A partialanswer to the raised question is given by the followingProposition 2. If Ŝ is the A(D)-hull of S, then the set bŜ ∩ D is thedisjoint 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 Shas 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

SEMI LOCAL LEVI-FLAT EXTENSIONS 25and 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 holomorphicdiscs. By “Kontinuitätssatz” we can easily see that, since the spheresS n ′ and S n ′′ were chosen to be disjoint, all the balls B n ′ and B n ′′ will alsobe disjoint. For each n =1, 2,... denote by M n the closed connectedcomponent of D(B n ′ ∪ B n) ′′ containing S on its boundary and byN n the closed connected component of bD (S n ′ ∪ S n) ′′ containing S.Then {M n } is a decreasing sequence of compact subsets of D, therefore⋂M = ∞ M n is also a compact subset of D. We claim that M = Ŝn=1and bM∩D is the disjoint union of holomorphic discs with boundaryin S.To show that Ŝ ⊂M we observe that, by the definition of M, foreach ε>0 there are balls B n ′ and B n ′′ contained in the ε-neighbourhoodU ε (M) ofM. By Lemma 1, we know that the set DŜ 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, dueto 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 ⊂ ∞ M n = M.n=1Now we prove that bM∩D is the disjoint union of holomorphic discswith boundary in S. In paricular, by virtue of the maximum modulusprinciple, this will imply the other inclusion M⊂Ŝ and, hence, thatM = Ŝ. Let Q beapointofbM ∩D. It follows then from thedefinition of M that either there is a sequence of points Q n ∈ B n ′ ora sequence of points Q n ∈ B n ′′ converging to Q. Since the argumentin both cases is the same, we assume that Q n ∈ B n. ′ Let D n be theholomorphic disc given by [Kr] which is contained in B n ′ and such thatQ n ∈ D n . Since the area of D n is less than the area of S n ′ = bB n, ′ andsince the spheres S n ′ converging to S can be chosen in such a way thatthe areas of S n ′ are uniformly bounded, we know by [Bi] that there isa subsequence {D nk } converging to a holomorphic disc D Q . It follows⋂then from the inclusion D nk ⊂M nk and the definition of M = ∞M nn=1that D Q ⊂M. Since D nk ⊂ B n ′ k, and since, by the argument above,B n ′ k∩M = ∅, we conclude that D Q does not intersect the interiorpart of M and, hence, D Q ⊂ bM ∩D. To show the unicity of theholomorphic disc contained in bM∩D and passing through Q we argueby contradiction assuming that there is another disc D ≠ D Q . SinceQ ∈ D ∩ D Q , and since the property of two holomorphic discs in C 2

26 NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIto have a nonempty intersection is stable under small perturbation,we conclude that D ∩ D nk ≠ ∅ for k big enough. Then the fact thatM∩B n ′ k= ∅, D nk ⊂ B n ′ kand D ⊂M gives the desired contradictionthereby proving the unicity of D Q . It is easy to see that the unicityof the holomorphic disc contained in bM and passing through a givenpoint of bM ∩D implies that the whole set bM ∩D is the disjointunion of holomorphic discs with boundary lying in S. This completesthe proof of Proposition 2. ✷Note, that if the domain D is bounded and diffeomorphic to a 4-ball, 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 ′ ) andE(bD ′′ ) are disjoint and Ŝ = D(E(bD′ ) ∪E(bD ′′ )). As we have alreadymentioned above, we do not know if the interior part of the set Ŝ =D(E(bD ′ ) ∪E(bD ′′ )) is empty. Surprisingly, for D being a rigid domainin C 2 this can occur. More precisely, there is an open smoothly boundeddomain 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 × Rv ) ∪E((bBU) × R v ) ) is nonempty. An example of such setU is presented below.Example 2. Consider the unit ball B in C z × R u . Then, accordingto our notations in Section 4, the covering model G of B is the unitdisc ∆ = {|z| < 1}⊂C z and for the functions h ± defining bB we haveh ± (z) =± √ 1 −|z| 2 .Consider the smooth 2-disc in bB with the projections p(U ± )ofthesets U ± = U ∩ bB ± on∆asinfig.1.

SEMI LOCAL LEVI-FLAT EXTENSIONS 27It follows then from our definition of the sets Σ and U ± (see Section4) and from the description of E(U) given by Proposition 1 that the setbE(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 theboundary data of F ± are also indicated) and the segment B∩{z = A 1 }.Similarly, the set U c = bBU is also a 2-disc with the projectionsp(U c ± ) of the sets U c ± = U c ∩ bB ± on ∆ as in fig. 4. The correspondingsets Σ c and U c ± are shown in figg. 5 and 6 as well as the boundarydata of the functions F c ± , whose graphs together with the segmentsB ∩{z = A i }, i =2, 3, 4, constitute the set bE(U c ) ∩ B. It is clearnow that, since the graphs Γ(F ± ) do not coincide with the graphsΓ(F c ± ), and since the sets E(U) and E(U c ) are disjoint, there is anopen subdomain Ω of B, bounded by the graphs Γ(F ± ), Γ(F c ± ) andthe segments B ∩{z = A i }, i =1, 2, 3, 4, such thatB × 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 Leviflatsurfaces: L 1 , which is constituted by the hypersurfaces Γ(F ± )×R vfoliated by holomorphic discs and the strip (B ∩{z = A 1 }) × R v , andL 2 , which is constituted by the hypersurfaces Γ(F c ± ) × R v foliated byholomorphic discs and the strips (B ∩{z = A i }) × R v , i =2, 3, 4,. BothLevi-flat hypersurfaces L 1 and L 2 have the same boundary bU × R v . ✷Finally, we formulate some questions which remains open.

28NIKOLAY SHCHERBINA AND GIUSEPPE TOMASSINIQuestion 1. Describe the set E(U) in the intrinsic terms of the spaceC z × R u (i.e. without using the domain G × R v ⊂ C 2 z,w and the hullŝK n of compacts K n exhausting U × R v ).Question 2. For each open set U ⊂ bG find the maximal open subsetE of G such that every function ϕ ∈ C(U) has a continuous Levi-flatextension to E.References[BG] E.BEDFORD and B.GAVEAU, Envelopes of holomorphy of certain 2-spheres in C 2 , Amer.J.Math.105 (1983), 975-1009.[BK] E.BEDFORD and W.KLINGENBERG, On the envelopes of holomorphyof 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.

SEMI LOCAL LEVI-FLAT EXTENSIONS 29[CS][El][Er][G][Ke][Ki][Kr][N][Sh][ST][St][TT][T]E.M.CHIRKA and N.V.SHCHERBINA, Pseudoconvexity of rigid domainsand foliations of hulls of graphs, Ann.Scuola Norm.Sup.Pisa Cl.Sci.(4),XXI (1995), 707-735.Y.ELIASHBERG, Filling by holomorphic discs and its applications, LondonMath.Soc.Lecture Note Ser., 151 (1991), 45-67.O.G.EROSHKIN, On a topological property of the boundary of an analyticsubset of a strictly pseudoconvex domain in C 2 , Mat.Zametki 49 (1991),149-151.T.W.GAMELIN, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs,N.J., 1969.N.KERZMAN, Hölder and L p estimates for solutions of ¯∂u = f in stronglypseudoconvex domains, Comm.Pure Appl.Math.24 (1971), 301-379.C.KISELMAN, The partial Legendre transformation for plurisubharmonicfunctions, Invent.Math.49 (1978), 137-148.N.G.KRU˘ZILIN, Two-dimensional spheres in the boundary of strictly pseudoconvexdomains in C 2 , Izv.Akad.Nauk SSSR Ser.Mat.55 (1991), 1194-1237.R.NARASIMHAN, The Levi problem for complex spaces. II, Math.Ann.146 (1962), 195-216.N.SHCHERBINA, On the polynomial hull of a graph, Indiana Univ.Math.J. 42 (1993), 477-503.N.SHCHERBINA and G.TOMASSINI, The Dirichlet problem for Leviflatgraphs over unbounded domains, Internat.Math.Res.Notices (1999),111-151.E.L.STOUT, Removable singularities for the boundary values of holomorphicfunctions, Several complex varuables (Stockholm, 1987/1988), 600-629,Math.Notes, 38, Princeton Univ.Press, Princeton, NJ, 1993.E.CASADIO TARABUSI and S.TRAPANI, Envelopes of holomorphy ofHartogs and circular domains, Pacific J.Math.149 (1991), 231-249.G.TOMASSINI, Sur les algèbres A 0 (D) et A ∞ (D) d’un domaine pseudoconvexeborné, Ann.Scuola Norm.Sup.Pisa Cl.Sci.(4), X (1983), 243-256.Nikolay ShcherbinaDepartment of MathematicsUniversity of Göteborg412 96 GöteborgSWEDENGiuseppe TomassiniScuola Normale SuperiorePiazza dei Cavalieri 756126 PisaITALY