Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type
aa r X i v : . [ m a t h . C V ] M a r Exposing points on the boundary ofa strictly pseudoconvex or a locallyconvexifiable domain of finite 1-type
K. Diederich, J. E. Fornæss and E. F. Wold ∗ September 27, 2018
Abstract : We show that for any bounded domain Ω ⊂ C n of1-type k which is locally convexifiable at p ∈ b Ω , having a Steinneighborhood basis, there is a biholomorphic map f : ¯Ω → C n suchthat f ( p ) is a global extreme point of type k for f (Ω) . In this paper we consider bounded locally convexifiable domains Ω in C n offinite 1-type whose closures ¯Ω admit a Stein neighborhood basis. Here the term"locally convexifiable near p ∈ b Ω " means that there are a neighborhood V of p and a one-to-one holomorphic map Φ : V → C n such that Φ(Ω ∩ V ) is convex.For the notion of finite type we refer to [2]. Strongly pseudoconvex domais areexamples of such domains. We will first prove the following: Theorem 1.1
Let Ω ⊂ C n be a bounded domain which is locally convexifiableand has finite type k near a point p ∈ b Ω . Assume further that b Ω is C ∞ -smooth near p , and that Ω has a Stein neighborhood basis. Then there exists aholomorphic embedding f : Ω → B nk , where B nk = { z ∈ C n : | z n | + k z ′ k k < } ,such that1. f ( p ) = e n = (0 , . . . , , , and2. { z ∈ Ω : f ( z ) ∈ b B nk } = { p } .If k = 1 , i.e. , if b Ω is strongly pseudoconvex near p , it is enough to assume that b Ω is C -smooth near p . Definition 1.2
Let Ω ⊂ C n be a domain and let p ∈ b Ω be a point. We saythat p is a globally exposed k -convex point if there exists an affine linear map f as in the previous theorem. ∗ Supported by the NFR grant 209751/F20 C ∞ family of local holomor-phic support functions ˆ S ( z, ζ ) ∈ C ∞ ( C n × ∂D ) for locally convexifiable domainsof finite type k with Stein neighborhood basis as explained in [4]. It has beenasked several times whether these support functions can always be chosen suchthat they are globally supporting for the given domains.However, it has to be asked in which way precisely this question should beanswered. As far as we can see, there are at least the following different possi-bilities, each of them leading to quite different answers:1. Our original support surfaces are defined only locally. The danger mightbe, for instance, that after a while they fall back into the inside of thedomain or, at least, become tangent at certain points, that are furtheraway. However, this danger can be avoided by applying a simple standard ∂ -argument to the defining functions of the support functions. Then weget new support functions which are well-defined in a possibly narrowStein neighborhood of Ω .2. Asking for more might mean that we really want globally defined supportsurfaces, i.e. , support surfaces which are closed smooth complex hyper-surfaces in C n , touching b Ω only from the outside at one distinguishedboundary point. It is clear that this requires a much stronger hypothe-sis on the domain. Namely, we will assume that the given domain has aRunge neighborhood basis and is locally convexifiable of finite type near .It is one of the main results of this article (Theorem 1.3) that such closedglobal support surfaces then always do exist. Under suitable regularityassumptions on b Ω (namely b Ω has to be C ∞ -smooth) smooth C ∞ -familiesof such supporting hypersurfaces do indeed exist (Theorem 1.4).In this part of the work we will prove the following statement: Theorem 1.3
Assume in addition to the hypotheses in Theorem 1.1 that Ω hasa Runge and Stein neighborhood. Then the map f can be chosen as a globalautomorphism of C n . A special case of this are convex domains of finite 1-type. Finally, in the case of bounded and smooth convex domains, we prove a versionof Theorem 1.1 with parameters:
Theorem 1.4
Let Ω ⊂ C n be a smooth and bounded convex domain of finitetype k . There exists a smooth family ψ ζ ∈ Aut hol C n , ζ ∈ b Ω , such that ψ ζ ( ζ ) is a globally exposed k -convex boundary point for the domain ψ ζ (Ω) . The structure of the article is as follows: In Section 2 we recall some localproperties of convexifiable domains due to the two first authors. In Section 3we prove Theorem 1.1. In Section 4 we prove Andersén-Lempert theorems with2arameters needed to prove Theorem 1.4, which we will do in Section 5. Finally,in Section 6, we give a brief sketch of how to prove Theorem 1.3 based on thearguments in Sections 3 and 5.
Let Ω be a bounded C ∞ -smooth domain in C n . In this section we recall themain facts about supporting hypersurfaces constructed in [4] For this we supposethat there is an open set V ⊂ C n such that b Ω ∩ V is convex. Near any point ζ ∈ b Ω ∩ V there is an open neighborhood V ζ of ζ , and a choice of a C ∞ -familyof coordinate changes { l ζ ( z ) : ζ ∈ b Ω ∩ V ζ } composed of a translation and aunitary transformation, such that, for each ζ ∈ b Ω ∩ V ζ , l ζ ( ζ ) = 0 and the unitoutward normal vector n ζ at ζ is turned by l ζ into the unit vector (1 , , . . . , .In particular, T C ζ b Ω becomes in the new coordinates ˜ z = l ζ ( z ) associated to ζ just { ˜ z = 0 } . The following is proved in [4]: Theorem 2.1
In the situation just described, assume that b Ω ∩ V is of finite1-type k , and let ˜ V ⊂⊂ V . Then there exists a function b S ( ζ, z ) ∈ C ∞ (( b Ω ∩ ˜ V ) × C n ) , and constants r, c > , such that the following holds: for any choiceof coordinate changes l ζ as above, the function S ( ζ, z ) := b S ( ζ, l − ζ ( z )) is equalto S ζ ( z ) = 3 z + Kz + g ζ ( z ′ ) , where ( z , z ′ ) are coordinates on C n , (1) and satisfies the estimate ReS ζ ( z ) ≤ − c ( | z | + k z ′ k k ) , (2) for all z ∈ B r ∩ l ζ (Ω) . Note that if the domain Ω is convex, we get a C ∞ -smooth function b S ( ζ, z ) on b Ω × C n . The proof of Theorem 1.1 is reduced to the two Lemmas in this section, Lemma3.1 and Lemma 3.3.We let e , . . . , e n be the standard basis for the complex vector space C n andput f j := i · e j so that e , f , . . . , e n , f n is a real basis. We denote the coordinateson C n by z j = x j − + ix j , we let C n denote the complex line C n = { z ∈ C n : z = . . . = z n − = 0 } and we let π n be the orthogonal projection to C n .Our proof uses a technique from [8] invented for exposing points on a borderedRiemann surface in order to produce a proper holomorphic embedding (see also[6] Sections 8.8 and 8.9). We suppose that Ω is convexifiable near some point p on its boundary. Then we get the following situation:3 emma 3.1 For any p ∈ b Ω there exists Φ ∈ Aut hol C n such that the followinghold1. Φ( p ) = 0 and T ( b Φ(Ω)) = { x n = 0 }
2. The outward normal to b Φ(Ω) at the origin is f n ,3. Near the origin we have that b Φ(Ω) is k -convex at the origin in the fol-lowing sense: The domain Φ(Ω) ⊂ { z ∈ C n : x n − f ( z ′ , x n − ) ≤ } with f ( z ′ , x n − ) ≥ c ( k z ′ k k + x n − ) , c > and4. Φ(Ω) ∩ { z ∈ C n : z = . . . = z n − = x n − = 0 , x n ≥ } = { } Definition 3.2
When condition (3) is satisfied near the origin we will refer tothe origin as a strictly k -convex boundary point. Proof: It follows by Corollary 2.4 in [4] that there exists an open neighborhood U p of p and an injective holomorphic map ψ : U p → C n such that ψ (Ω ∩ U p ) satisfies (1)-(3). Choosing an appropriate neighborhood V p ⊂ U p of p , it followsthat ψ is approximable by automorphisms φ of C n uniformly on V p (see Section4) and that ψ (Ω) is strictly k-convex near Φ( p ) if φ is close enough to ψ .We proceed to achieve (4). Let Γ := { z ∈ C n : z = . . . = z n − = x n − = 0 , x n ≥ } , (3)and let Γ := { z ∈ C n : z = . . . = z n − = x n − = 0 , ≤ x n ≤ } . (4)Choose an R > such that Ω ⊂ B nR . By [7] there exists ψ ∈ Aut hol ( C n ) such that ψ ( z ) = z + O ( k z k k +1 ) as z → , such that ψ (Γ ) ∩ Ω = { } , andsuch that ψ ( q ) ∈ C n \ B nR ,where q denotes the endpoint of Γ other than theorigin. Consider the set ψ − ( B nR ) ∩ Γ ⊂ Γ \{ q } . Since ψ − ( B nR ) is polyno-mially convex we have that ( C n − × { i } ) \ ψ − ( B nR ) is connected, and so usingWeierstrass approximation theorem, we may construct a holomorphic shear map ψ ( z ) = ( z + f ( z ) , . . . , z n − + f n − ( z n ) , z n ) such that ψ is close to the identityon Γ , tangent to the identity to order 2k+1 at the origin, and therefore notdestroying strict k-convexity at 0, and such that ψ ((Γ \ Γ ) ∩ ψ − ( B nR ) = ∅ . So ( ψ ◦ ψ )(Γ) ∩ Ω = ∅ , and we set Φ = ( ψ ◦ ψ ) − . Lemma 3.3
Let W ⊂ C n be a bounded domain with ∈ bW and assume thatthe following hold (i) W has a Stein neighborhood basis, (ii) W is strictly k-convex near the origin, (iii) W ∩ Γ = { } with Γ defined as in the proof of Lemma 3.1. hen for any open set ˜ V containing Γ and any small enough open set V ⊂ ˜ V containing the origin, there exist a sequence of holomorphic embeddings f j : W → C n such that the following holds ( f j → id uniformly on W \ V as j → ∞ ( f j ( V ) ⊂ ˜ V for all j , ( f j (0) = f n for all j , ( Im( π n ( f j ( z ))) < for all z ∈ ( W ∩ V ) \ { } and ( f j ( W ) is strictly k-convex at f j (0) . Proof: We let ˜Ω ⊂ C n denote the domain ˜Ω := { z n ∈ ∆ ε : x n < f (0 , . . . , , x n − } ) .For some small δ > we define the following sets: A := { z ∈ W ∩ B nε : x n ≥ − δ and B := { z ∈ W ∩ B nε : x n ≤ − δ } ∪ W \ B nε . Then A \ B ∩ B \ A = ∅ (if δ is small) and by Theorem 4.1 in [5], for any open set ˜ C containing theset C := A ∩ B there exist open sets A ′ , B ′ , C ′ with A ⊂ A ′ , B ⊂ B ′ and C ⊂ C ′ ⊂ A ′ ∩ B ′ ⊂ ˜ C , such that if γ : ˜ C → C n is injective holomorphic,and sufficiently close to the identity, then there exist holomorphic injections α : A ′ → C n , β : B ′ → C n , uniformly close to the identity on their respectivedomains (depending on γ ), and such that γ = β ◦ α − on C ′ . (5)(This can also be found in Theorem 8.7.2, page 359 in [6].) Choose a simplyconnected smooth domain U ⊂ C n with π n ( A ) ⊂ U and such that near theorigin U = { z ∈ C n : x n < } . For j ∈ N let l j denote the line segment l j = { z n ∈ C n : x n − = 0 , ≤ x n ≤ /j } . For each j it follows from Mergelyan’sTheorem that we may choose injective holomorphic maps σ j : U ∪ l j → C n such that σ j approximately stretches l j to cover Γ such that σ j ( z ) = (1 − /j ) i + z + O ( | z − i/j | ) k +1 and such that σ j → id on U as j → ∞ . For each j let U j be a domain obtained from U by adding a strip around l j of width lessthan /j which is then smoothened and made strictly convex at the end point l j . U j should lie inside where σ j is injective holomorphic, and be chosen suchthat σ j ( U j ) is strictly convex near the end point of σ j ( l j ) = f n and such that Im( σ j ( z n )) < for all z ∈ U j \ j f n . Let ψ j be a holomorphic diffeomorphismfrom U to U j such that ψ j (0) = ij , ψ j → id uniformly on U . (See Goluzin, [10],Theorem 2, p. 59.) Let φ j = σ j ◦ ψ j and let γ j be an extension of φ j to A. Then Im (Π n ( γ j ( z ))) < for all z ∈ A \ { } . It is not hard to see that γ j ( A ) is strictlyk-convex near f n and γ j → id on a neighborhood of C . We get splittings γ j ◦ α j = β j (6)as explained above. If j is large enough, we get that (7) defines an injectiveholomorphic map f j on Ω , and if α j is close enough to the identity, since α j canbe assumed to vanishes to order k + 1 at the origin, we get that Im( f j ( z )) < for all z ∈ A \ { } and such that f j ( A ) is strictly k-convex at f j (0) .5 Andersén-Lempert with parameters in a smoothmanifold, and approximation with jet interpolation.
A parameter version of the Andersen-Lempert theorem [1] for holomorphic pa-rameters was proved by Kutzschebauch[11]. Jet interpolation results withoutparameters have been proved by Forstnerič [9] and Weickert [13] (see also sec-tions 4.9 and 4.15 in [6]). For a smooth manifold M we let ( ζ, z ) denote thecoordinates on M × C n . For any ζ ∈ M we denote by C nζ the slice { ζ } × C n ,and for any subset Σ ⊂ M × C n we let Σ ζ denote the slice Σ ζ := C nζ ∩ Σ . Theorem 4.1
Let M be a compact smooth manifold and let Ω ⊂ M × C n be adomain, n ≥ . Let K ⊂ Ω be a compact set, and let φ : [0 , × Ω → M × C n be a C -smooth map such that, writing φ ( t, ζ, z ) = φ t ( ζ, z ) , the following hold(1) φ t ( ζ, z ) = ( ζ, ϕ t ( ζ, z )) = ( ζ, φ t,ζ ( z )) ,(2) φ t,ζ : Ω ζ → C nζ is injective holomorphic, and(3) K t,ζ := φ t,ζ ( K ζ ) varies continuously with ( t, ζ ) and is polynomially convexfor all t ∈ [0 , , ζ ∈ M .Then φ is uniformly approximable on K by a smooth family ψ ( ζ, z ) with ψ ζ ∈ Aut hol C nζ if (and only if ) φ is approximable by such a family. Moreover, if(1)–(3) hold and if a ( ζ ) ∈ K ◦ ζ is a smoothly parametrized family of points, andif d ∈ N , we may additionally achieve that(4) φ ,ζ ( z ) − ψ ζ ( z ) = O ( k z − a ( ζ ) k d ) , as z → a ( ζ ) . Proof: We give a sketch of the proof of the first claim; the point is just to verifythat the non-parametric proof goes through without change with parameters.The assumption that φ is approximable allows us to assume φ = id . Definefirst a parametrized vector field X t,ζ ( φ t,ζ ( z )) := ddt φ t,ζ ( z ) . (7)Then X t,ζ is an inhomogeneous vector field, holomorphic in z , whose flow is φ t,ζ ( z ) . For each t let ϕ st,ζ denote the time- s flow of the homogenous vector field X t,ζ where t is fixed. It is well known that there is a partitioning [ j/n, ( j +1) /n ] , j = 0 , ..., n − of [0 , , such that the composition ϕ /n ( n − /n,ζ ) ◦ · · · ◦ ϕ /n ,ζ (8)approximates φ ζ, on K . So the problem is reduced to approximating the flow ϕ ζ of a homogenous vector field X ζ on a family K ζ .Next, by assumption (3) and approximation, we may assume that X ζ is apolynomial vector field X ζ ( z ) = N X j =1 g j ( ζ ) X j ( z ) , (9)6ith coefficients g j in E ( M ) ; this can be obtained by gluing a fiberwise Runge-approximation using a partition of unity on M . Now the main point of Andersén-Lempert Theory in C n is that any m -homogenous polynomial vector field V m isa sum of completely integrable vector fields (see e.g. [6], Lemma 4.9.5): V m ( z ) = r X i =1 c i λ i ( z ) m · v i + d i λ i ( z ) m − h z, v i i · v i , (10)with c i , d i ∈ C , v i ∈ C n and λ i ∈ ( C n ) ∗ with λ i ( v i ) = 0 . The flows of these twotypes of vector fields are z f t,j z + t · c i λ i ( z ) m · v i and z g t,j z + ( e td i λ i ( z ) m − h z, v i i · v i . (11)Applying this to each of the vector fields X j ( z ) in (9) we get that X ζ ( z ) = ˜ N X j =1 ˜ g j ( ζ ) · ˜ X j ( z ) , (12)where each ˜ X j is completely integrable with flow ψ sj , and so X ζ is a sum ofcompletely integrable fields with flows ψ sζ,j = ψ g ( ζ ) · sj . Finally the sequence ( ψ /nζ,N ◦ · · · ◦ ψ /nζ, ) n (13)converges uniformly to ϕ ζ as n → ∞ .Finally we consider (4). We will correct the initial approximation at a ( ζ ) and by translation we may assume that a ( ζ ) = 0 for all ζ , and that both φ and ψ fix the origin. Define J d − ( ζ ) to be the ( d − -jet of ψ − ζ ◦ φ ,ζ .It is easy to see that we may assume that J d − ( z ) = id + h.o.t , and by theCauchy estimates we may assume that J d − ( ζ ) is arbitrarily close to the identitymap. We will correct ψ ζ inductively, and our induction assumption is that J d − ( ζ ) = O ( k z k m ) , ≤ m ≤ d − .Using (10) we fix an expansion z α · e j = s α,j ( z ) := r X i =1 c α,ji λ α,ji ( z ) m · v α,ji + d α,ji λ α,ji ( z ) m − h z, v α,ji i · v i (14)for each multi-index | α | = m and j = 1 , ..., n . Now expand the m -homogenouspart J d − ,m of J d − using (14) J d − ,m ( ζ ) = X | α | = m, ≤ j ≤ n h α,j ( ζ ) · s α,j ( z ) . (15)It is easy to see that the composition Φ m of all automorphisms z z + h α,j ( ζ ) · c α,ji λ α,ji ( z ) m · v α,ji (16)7nd z z + ( e h α,j ( ζ ) d α,ji λ α,ji ( z ) m − h z, v α,ji i · v α,ji . (17)matches J d − ,m to order m , and we may assume that Φ m is as close to theidentity as we like on a compact set since all the h α,j ’s can be assumed to be assmall as we like. It follows that the map ψ ζ ◦ Φ m is a small perturbation of ψ ζ which matches φ ,ζ to order m . The induction step is complete. Remark 4.2
For a more detailed explanation of jet-completion (without param-eters) the reader can consult [6] page 154–158.
Theorem 5.1
Let Ω ⊂ C n be a smooth and bounded convex domain of finitetype k . There exists a smooth parameter family ψ ζ ∈ Aut hol C n , ζ ∈ b Ω , suchthat ψ ζ ( ζ ) is a globally exposed k -convex boundary point for the domain ψ ζ (Ω) . Proof: By [4] there exist r, c > and a smooth parameter family ψ ζ ( z ) defined on { ( ζ, z ) : k z − ζ k < r } (18)such that ψ ( ζ, · ) is injective holomorphic for all ζ and the following holds forall ζ (see Section 2): let n ζ denote the outward pointing unit normal vector to b Ω at ζ , let l ζ be a composition of a translation and a unitary transformationsuch that l ζ ( ζ ) = 0 and such that n ζ is sent to the vector (1 , , ..., . Then ˜ ψ ζ ( z ) := ψ ζ ◦ l − ζ is of the form ( S ζ ( z ) , z , ..., z n ) , and S satisfies S ζ ( z ) = 3 z + Kz + g ζ ( z ′ ) , z = ( z , z ′ ) , (19)(See Section 2.) Moreover, we have that Re ( S ζ ( z )) ≤ − c · (cid:16) | z | + k z ′ k k (cid:17) , z ∈ B r ( ζ ) ∩ Ω , (20)where the constant c > does not depend on ζ .Our first step is to change the maps ψ ζ conveniently on the normals n ζ , andthen approximate the changed maps by a family of holomorphic automorphisms.Set Γ := { z ∈ C n : 0 ≤ x ≤ , x = z = ... = z n = 0 } , and let h denote themap h ( z ) = 3 z + Kz near Γ . By changing h smoothly, then finding a smoothhomotopy of maps, and finally applying Mergelyan’s Theorem with parameters,we find δ > and a smooth map ˜ h : [0 , × Γ ( δ ) → C , such that the following hold(1) ˜ h ( z ) = 3 z ,(2) ˜ h t ( · ) is injective holomorphic for each t ∈ [0 , ,83) ˜ h ( z ) ≈ h ( z ) on B δ (0) and (˜ h − h )( z ) = O ( | z | k +1 ) as z → ,(4) ˜ h ( z ) ≈ z on B δ (1) and (˜ h − z ) = O ( | z − | k +1 ) as z → ,(5) ˜ h − (3Γ ) ≈ Γ .We define a homotopy modification b ψ ζ,t ( z ) of ψ ζ by setting b S ζ,t ( z ) := ˜ h t ( z ) + t · g ζ ( z ′ ) on Γ ( δ ) . (21)in local coordinates.Let b ( ζ ) denote the end point of n ζ other than ζ , and note that by Stolzenberg[12] we may assume, by possibly having to decrease δ , that b K ζ,t := b ψ ζ,t ( B δ ( ζ ) ∪ n ζ ∪ B δ ( b ( ζ )) is polynomially convex for all ζ . By Theorem 4.1 and its proof there existfamilies G ζ , H ζ ∈ Aut hol C n such that the following holds(6) G ζ ≈ b ψ ζ, on B δ ( ζ ) , and ( G ζ − b ψ ζ )( z ) = O ( k z − ζ k k +1 ) as z → ζ ,(7) H ζ ≈ b ψ − ζ, on B δ (0) ∪ n ζ ∪ B δ (3 b ( ζ )) ,(8) ( H ζ − b ψ − ζ, )( z ) = O ( k z − b ( ζ ) k k +1 ) as z → b ( ζ ) , and(9) H ζ ◦ G ζ ≈ id on Ω .Next we construct a continuous parameter family of exposing maps f ζ as inLemma 3.3, where each f ζ wraps the boundary at G ζ ( ζ ) around the normal n ζ . The composition H ζ ◦ f ζ ◦ G ζ will globally expose the point ζ k -convexly.We will then change f ζ to depend smoothly on ζ , and in a final step we willapproximate the family f ζ by a smooth family of automorphisms.Choose a strictly pseudoconvex neighborhood Ω ′ of Ω close to Ω and let ρ be a smooth strictly plurisubharmonic defining function for Ω ′ near b Ω ′ . For < r << we let Ω ′ ( r ) := { z : ρ ( z ) < r } . For < σ << we defineCartan pairs ˜ A ζ ( r ) := Ω ′ ( r ) ∩ B σ ( ζ ) , ˜ B ζ ( r ) := Ω ′ ( r ) \ B σ/ ( ζ ) . Set A ζ ( r ) := G ζ ( ˜ A ζ ( r )) , B ζ ( r ) := G ζ ( ˜ B ζ ( r )) .Let γ j be the sequence of locally exposing maps from the proof of Lemma 3.3.Since the maps only depend on the normal coordinate, the map ˜ γ j,ζ := l − ζ ◦ γ j ◦ l ζ is a well defined family of locally exposing maps for Ω , and ˜ γ j,ζ → id uniformlyon C ζ ( r ) := A ζ ( r ) ∩ B ζ ( r ) for small enough r independently of ζ . To globalizethese locally defined maps we use the following parametric version of Theorem8.7.2 in [6]. Lemma 5.2 If r is small enough and µ > there exist r < r and ǫ > suchthat the following holds: for any family γ ζ : C ζ ( r ) → C n of holomorphic mapswith k γ ζ − id k C ζ ( r ) < ǫ , continuous in ζ , there exist injective holomorphic maps α ζ : A ζ ( r ) → C n , β ζ : B ζ ( r ) → C n , continuous in ζ , such that γ ζ = β ζ ◦ α − ζ , k α ζ − id k A ζ ( r ) < µ, k β ζ − id k B ζ ( r ) < µ. (22) Moreover, we may achieve that ( α ζ − id )( z ) = O ( k z − ζ k k +1 ) as z → ζ . ∂ -equation which is continuous with parameters, andone can multiply by powers of S ζ to get exact jet interpolation.So if j is chosen large enough we get that the family f ζ defined as f ζ := γ ζ,j ◦ α j,ζ on A ζ ( r ) and f ζ := γ ζ,j ◦ β ζ,j on B ζ ( r ) , is a family of injectiveholomorphic maps ˜ γ ζ : G ζ (Ω ′ ( r )) → C n exposing the point ζ k -convexly.By (5) and (8) the family H ζ ◦ f ζ ◦ G ζ is a continuous family of holomorphicinjections on Ω ′ ( r ) , globally exposing the point ζ for the domain Ω .Next we approximate f ζ by a smooth family of exposing maps. This is doneusing a partition of unity on b Ω . Note first that although f ζ is only continuous in ζ , the k -jet at ζ , J ( ζ ) , is smooth in ζ ; this is because α j,ζ vanishes to order k at ζ . Let ( U j , α j ) , j = 1 , ..., m , be a partition of unity on b Ω with a point a j ∈ U j forall j . For each j write f a j ( z ) = z + g j ( z ) . We set ˜ f ζ ( z ) := z + P mj =1 α j ( ζ ) g j ( z ) .By choosing the covering fine enough we may achieve that ˜ f ζ is as close to f ζ as we like on Ω , and also that the k -jet of ˜ f ζ at ζ is as close to that of f ζ aswe like. So using the argument in the proof of Theorem 4.1 we can correct ˜ f ζ so that its k -jet at ζ matches that of f ζ exactly.Finally we need to approximate the family ˜ f ζ by a family of automorphisms.We may assume that ∈ Ω , G ζ (0) = 0 , and that f ζ (0) = 0 for all ζ . Set ϕ t ( ζ, z ) := G ζ ( 1 t ˜ f ζ ( t · G − ζ ( z ))) . (23)We may assume that ˜ f ζ ( G ζ (Ω)) is polynomially convex for all ζ ∈ b Ω . Inthat case it follows that there exists some s > such that ϕ t,ζ ( G ζ ( s Ω)) ispolynomially convex for all t, ζ , and so approximation follows by Theorem 4.1.It is enough to show that f ζ ( G ζ (Ω)) is polynomially convex.Fix ζ ∈ b Ω . By Stolzenberg [12] we have that G ζ (Ω) ∪ n ζ is polynomiallyconvex. Let W ζ be a Runge neighborhood of K ζ := G ζ (Ω) ∪ n ζ , very close to K ζ . Consider a point b ∈ b Ω ∩ B ζ (0) . If W ζ is close enough to K ζ , and if β ζ isclose enough to the identity, then the locally defined function e C · S b ( β − ζ ( z )) for C >> may be approximately globalized to W ζ , separating points on β ζ ( n b ) close to β ζ ( b ) from f ζ ( G ζ (Ω)) as long as f ζ is chosen such that f ζ (Ω) ⊂ W ζ . Itfollows that cl [ \ f ζ ( G ζ (Ω)) \ f ζ ( G ζ (Ω))] ∩ f ζ ( G ζ (Ω)) ⊂ f ζ ( A ζ (0)) . (24)Hence by Rossi’s local maximum principle \ f ζ ( G ζ (Ω)) = f ζ ( G ζ (Ω)) ∪ \ [ f ζ ( A ζ (0)) ∩ G ζ (Ω)] . (25)But f − ζ is approximable by entire maps on f ζ ( A ζ (0)) , and so f ζ ( G ζ (Ω)) ispolynomially convex. The proof of Theorem 1.3 is almost the same as that of Theorem 1.1, exceptthat we need to make sure that the exposing maps f j are approximable by10olomorphic automorphisms. To see why this is so, note first that each γ j maybe connected to the identity map by an isotopy which is uniformly close tothe identity on C . The Cartan type splitting with parameters then allows usto construct each f j as the time-1 map of an isotopy f j,t with f j, = id (thisargument allows us to avoid the usual assumption in Andersén-Lempert theorythat Ω is star shaped). This isotopy is only C but we can obtain a smoothisotopy by gluing as before. The same argument as in the previous sectiontells us that we may assume that f t,j (Ω) is polynomially convex for all t if j issufficiently large, and so we may approximate by automorphisms. Acknowledgement 6.1
The authors would like to thank the referee for a verycarefull reading, thus helping us to improve greatly the exposition of the paper.
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