[.] [.]
Runge tubes in Stein manifolds with
the density property
Franc Forstnerič and Erlend F. Wold
Abstract In this paper we give a very simple proof of the existence and plenitude of Runge tubes in and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold of codimension at least two in a complex Euclidean space , the normal bundle of in admits a holomorphic embedding onto a Runge domain in which agrees with the inclusion map on the zero section.
Keywords Runge tube, holomorphic automorphism, Stein manifold, affine algebraic manifold, density property
MSC (2010): 32E10; 32E30; 32H02; 32M17; 14R10
1. Introduction
It was an open question for a long time whether it is possible to embed as a Runge domain in . (Here, is the complex plane and .) Such domains have been called Runge cylinders in . The question arose in connection with the classification of Fatoucomponents for Hénon maps by E. Bedford and J. Smillie in 1991, [4]. This problem has recently been solved in the affirmative by F. Bracci, J. Raissy and B. Stensønes [5] who obtained a Runge embedding of in as the basin of attraction of a (nonpolynomial) holomorphic automorphism of at a parabolic fixed point.
The purpose of this note is to give a very simple proof of the existence of Runge cylinders, and furthermore of the existence of an abundance of Runge tube domains in all Stein manifolds with the density property. Although our proof is completely different from that in [5], both proofs depend crucially on AndersénLempert theory (see [11, Chapter 4]).
Recall (see D. Varolin [18, 19] or [11, Definition 4.10.1]) that a complex manifold has the density property if every holomorphic vector field on is a uniform limit on compacts of finite sums of complete holomorphic vector fields on . In particular, Euclidean spaces of dimension have the density propery by E. Andersén and L. Lempert [1].
The following is our first main result.
Theorem 1.1.
Let and be Stein manifolds with , and assume that has the density property. Suppose that is a holomorphic embedding with convex image (this holds in particular if is proper), and let denote the normal bundle associated to . Then, is approximable uniformly on compacts in by holomorphic embeddings of into whose images are Runge domains.
Recall that a locally closed subset of a complex manifold is said to be convex if for every compact set , its convex hull
(1.1) 
is compact and contained in .
To get a Runge embedding of into from Theorem 1.1, one embeds onto the curve and notes that any vector bundle over (and in fact over any open Riemann surface) is trivial by Oka’s theorem [16]. (See also [11, Sect. 5.2].)
The authors together with R. Andrist and T. Ritter proved in [3, 2] that every Stein manifold embeds properly holomorphically into any Stein manifold with the density property satisfying . Every open Riemann surface embeds properly holomorphically into , and a plenitude of them embed properly into ; see [10] and [11, Sects. 9.109.11] for a discussion. By Theorem 1.1, any such embedding can be approximated by a Runge embedding of the normal bundle of in . This provides a huge variety of nontrivial Runge tubes in any Stein manifold with the density property. In particular, we have the following corollary to Theorem 1.1.
Corollary 1.2 (Runge tubes over open Riemann surfaces).
If is an open Riemann surface which admits a proper holomorphic embedding into , then admits a Runge embedding into . For every open Riemann surface and every , admits a Runge embedding into , and into any Stein manifold with the density property.
We wish to point out that there is a big list of Stein manifolds, and in particular of affine algebraic manifolds, which are known to have the density property. The reader may wish to consult the list of examples in [2], as well as the recent surveys of Kaliman and Kutzschebauch [15] and of the first named author [11, Sect. 4.10].
The Runge embeddings of the normal bundle in Theorem 1.1 need not agree with the embedding on the zero section of (which we identify with ). However, we can ensure this additional condition for algebraic embeddings of codimension at least 2 into . Here is the precise result.
Theorem 1.3.
Let be a Stein manifold and be proper holomorphic embedding onto an algebraic submanifold . If , then extends to a holomorphic Runge embedding of the normal bundle of .
By a theorem of Docquier and Grauert [6] (see also [11, Theorem 3.3.3]), every proper holomorphic embedding of a Stein manifold into a complex manifold extends to an embedding of a neighborhood of the zero section in the normal bundle of . What we find especially interesting in the context of Theorem 1.3 is that one can embed the entire normal bundle of as a Runge domain in .
The proof of Theorem 1.3 is similar to that of Theorem 1.1. It uses the result of Kaliman and Kutzschebauch [14, Theorem 6] that the Lie algebra of algebraic vector fields on vanishing on an algebraic submanifold of codimension at least two enjoys the algebraic density property. It follows that flows of such vector fields can be approximated by automorphisms of fixing the submanifold pointwise.
Corollary 1.4.
Let be an affine algebraic curve. Every proper algebraic embedding for extends to a holomorphic embedding onto a Runge domain in .
Our proof of Theorem 1.3 gives embeddings of the normal bundle which are holomorphic but not necessarily algebraic. In fact, when , the embedding in Corollary 1.4 cannot be chosen algebraic in general (even without the Runge condition on the image domain ). The reason is that there are affine algebraic curves in whose normal bundle is not algebraically trivial (although it is holomorphically trivial by Oka’s theorem). See Forster and Ohsawa [7] for a discussion and further references on this subject. However, we do not know the answer to the following question.
Problem 1.5.
Let be a smooth algebraic submanifold of codimension at least two, and let denote the algebraic normal bundle of the embedding. Is there an algebraic Runge embedding which agrees with the given embedding on then zero section of ?
2. Proof of Theorems 1.1 and 1.3
We begin by recalling some basic facts from the theory of Stein manifolds (see e.g. Gunning and Rossi [12] or Hörmander [13]) and explaining the setup.
A domain in a complex manifold is said to be Runge in if is a dense subset of . If both and are Stein, this holds if and only if for every compact subset we have that . In particular, a domain in a Stein manifold which is exhausted by compact convex sets is Runge in .
A holomorphic embedding of a complex manifold into a complex manifold is said to be Runge if the image is an convex subset of , i.e., it is exhausted by compact convex subsets. If and are Stein, then every proper holomorphic embedding is Runge.
Assume that is a holomorphic vector bundle over a Stein manifold . The total space is then also a Stein manifold. We shall write elements of in the form where , identifying with the zero section of . For any there is a holomorphic fibre preserving map
(2.1) 
Clearly, is a holomorphic automorphism of for every .
A subset is called radial if holds for every .
The following lemma provides the induction step in the proof of Theorem 1.1.
Lemma 2.1.
Assume that is a Stein manifold, is a holomorphic vector bundle, are compact radial convex subsets of , is an open set containing , is a Stein manifold with the density property such that , and is a holomorphic embedding such that is a Runge embedding and is convex. Then, can be approximated as closely as desired uniformly on by a holomorphic embedding of a domain with such that is a Runge embedding and the sets and are convex.
If, in addition to the hypotheses above, with and is a closed algebraic submanifold of , then the approximating embedding can be chosen to agree with on .
The conditions imply that is the normal bundle of the embedding .
Proof.
We identify with the zero section of . Choose a compact convex subset such that . Since the embedding is Runge, the image is convex. Pick a compact convex neighborhood of (such exists since an convex set has a basis of compact convex neighborhoods). Thus, for a compact set with .
Let be defined by (2.1). Since and is a neighborhood of in , we can choose small enough such that . Since is convex and , the set is convex, and hence afortiori convex. Since is a biholomorphism, it follows that the set is convex, and hence also convex (since is convex).
Consider the isotopy of injective holomorphic maps for , defined on an open neighborhood of in by the condition
(2.2) 
Note that the following hold:

is the identity map, and

for every the compact set is convex.
Condition (b) holds because is clearly convex, so is convex and hence convex (since is convex).
By the AndersénLempertForstneričRosayVarolin theorem (see [9, Theorem 2.1] for the case and [11, Theorem 4.10.5] for the general case), can be approximated uniformly on a neighborhood of by holomorphic automorphisms .
Since by the choice of , there is an open neighborhood of such that . We claim that the holomorphic embedding
(2.3) 
satisfies the lemma. Indeed, since the sets and are convex and is an automorphism of , the sets and are also convex. Furthermore, is a Runge embedding since is. Finally, on the set we have in view of (2.2) that
Since is close to the identity on by the choice of , it follows that is close to on . This proves the first part of the lemma.
Assume now that and that is a closed algebraic submanifold, where . By Kaliman and Kutzschebauch [14, Theorem 6], the Lie algebra of algebraic vector fields on vanishing on enjoys the algebraic density property. (This means that every algebraic vector field on vanishing on can be expressed by sums and commutators of complete algebraic vector fields on vanishing on . Indeed, one may use shear vector fields vanishing on .) This implies (see [11, Proposition 4.10.4]) that the flow of any algebraic vector field vanishing on can be approximated on each compact polynomially convex subset by holomorphic automorphisms of fixing pointwise
Note that, up to a change of the parameter, the isotopy (2.1) is the flow of a holomorphic vector field on , tangent to the fibres of the projection and vanishing on the zero section . Hence, the isotopy defined by (2.2) is also the flow of a holomorphic vector field on a neighborhood of in that vanishes on the algebraic submanifold . (Indeed, is the pushforward of by the embedding .) By Serre’s Theorems A and B [17] we can approximate as closely as desired on a neighborhood of by an algebraic vector field vanishing on . By what has been said above, this shows that the map can be approximated uniformly on a neighborhood of by holomorphic automorphisms such that for all . The proof is now completed just as before. In particular, we see from (2.3) that the embedding agrees with on . ∎
Proof of Theorem 1.1. Pick an exhaustion by compact radial convex sets. In fact, we may choose each of the form
where is a strongy plurisubharmonic exhaustion function on and is a suitably chosen hermitian metric on . Let be a holomorphic Runge embedding. By a theorem of Docquier and Grauert (see [11, Theorem 3.3.3]) there is a neighbourhood of the zero section such that extends to a holomorphic embedding . Set . By applying Lemma 2.1 inductively, we find a sequence of open neighbourhoods of and holomorphic embeddings satisfying the following conditions for every :

the compact sets and are convex,

the embedding is Runge, and

approximates as closely as desired on .
If the approximations are close enough, the sequence converges uniformly on compacts in to a holomorphic embedding . Since convexity of a compact set in a Stein manifold is a stable property for compact strongly pseudoconvex domains [8], and every compact convex set can be approximated from the outside by such domains, it follows that the image of each remains convex in the limit provided that all approximations were close enough. Hence, is a Runge domain in . ∎
Proof of Theorem 1.3. We follow the proof of Theorem 1.1. By the second part of Lemma 2.1, the sequence of embeddings can now be chosen such that, in addition to the above, we have that holds for all . This ensures that the limit embedding also satisfies . ∎
Acknowledgements
F. Forstnerič is supported by the research program P10291 and grant J17256 from ARRS, Republic of Slovenia. E. F. Wold is supported by the RCN grant 240569, Norway.
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Franc Forstnerič
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia
Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
email:
Erlend F. Wold
Department of Mathematics, University of Oslo, Postboks 1053 Blindern, NO0316 Oslo, Norway
email: