119298244
Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey
In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most of them pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map \(X\rightarrow Y\) between Stein manifolds is homotopic to a proper holomorphic embedding provided that \(\dim Y>2\dim X\) and we allow a homotopic deformation of the Stein structure on X.
124035982
MAPPINGS OF PARTIALLY ANALYTIC SPACES.
1. Introduction. We study a mapping f of a partially analytic space K into n-dimensional complex affine space Cn. A partially analytic space K is a Hausdorff topological space which is a countable union of conditionally compact open sets, which is locally connected, and which is endowed, at least partially, with an analytic structure. This means that a countable family {Ma} of subsets of K is given, each with the structure of a complex analytic manifold. An algebra A of continuous functions on K, called the analytic functions on K, is also given, with the property that each function in A is analytic on each Ml,a. The algebra A is closed in the topology of uniform coiivergence on compact subsets of K. The coordinates of the mapping f are required to lie in A. The purpose of this paper is to show that for certain spaces K the mapping f can be chosen to have certain properties. For instance, sufficient conditions are given foi there to exist an f which is proper, or which separates points, or which has a certain rank at certain points. Since the
117402231
Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis
Preliminaries. - Stein Manifolds. - Stein Neighborhoods and Holomorphic Approximation. - Automorphisms of Complex Euclidean Spaces. - Oka Manifolds. - Elliptic Complex Geometry and Oka Principle. - Applications. - Embeddings, Immersions and Submersions.- Topological Methods in Stein Geometry. - References. - Index.
118894044
Theory Of Stein Spaces
A. Sheaf Theory.- B. Cohomology Theory.- I. Coherence Theory for Finite Holomorphic Maps.- II. Differential Forms and Dolbeault Theory.- III. Theorems A and B for Compact Blocks ?m.- IV. Stein Spaces.- V. Applications of Theorems A and B.- VI. The Finiteness Theorem.- VII. Compact Riemann Surfaces.- Table of Symbols.- Addendum.- Errors and Misprints.
118335201
Analytic Functions of Several Complex Variables
The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.
122821814
An introduction to complex analysis in several variables
I. Analytic Functions of One Complex Variable. II. Elementary Properties of Functions of Several Complex Variables. III. Applications to Commutative Banach Algebras. IV. L2 Estimates and Existence Theorems for the Operator. V. Stein Manifolds. VI. Local Properties of Analytic Functions. VII. Coherent Analytic Sheaves on Stein Manifolds. Bibliography. Index.
59421037
Complete complex hypersurfaces in the ball come in foliations.
In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. 
Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q\in\{1,\ldots,n-1\}$.
117940141
Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for $n\ge 5$ we find embeddings with these properties.
119129424
Proper holomorphic embeddings into stein manifolds with the density property
We prove that a Stein manifold of dimension d admits a proper holomorphic embedding into any Stein manifold of dimension at least 2d + 1 satisfying the holomorphic density property. This generalizes classical theorems of Remmert, Bishop and Narasimhan, pertaining to embeddings into complex euclidean spaces, as well as several other recent results.
119155995
Proper holomorphic immersions into Stein manifolds with the density property
In this paper we prove that every Stein manifold S admits a proper holomorphic immersion into any Stein manifold of dimension 2 dimS enjoying the density property or the volume density property.
53136158
The density property for complex manifolds and geometric structures
Definition Let g be a Lie algebra of holomorphic vector fields. We say that g has the density property if the Lie subalgebra of g generated by the complete vector fields in g is dense in g. When the Lie algebra XO(M) of all holomorphic vector fields on a complex manifold M has the density property, we say that M has the density property. If (M,ω) is a calibrated complex manifold, i.e., ω is a nondegenerate holomorphic n-form (n = dimCM) on M , we say that (M,ω) has the volume density property if the Lie algebra XO(M,ω) of all holomorphic vector fields with vanishing ω-divergence has the density property.
53136158
The density property for complex manifolds and geometric structures
Definition Let g be a Lie algebra of holomorphic vector fields. We say that g has the density property if the Lie subalgebra of g generated by the complete vector fields in g is dense in g. When the Lie algebra XO(M) of all holomorphic vector fields on a complex manifold M has the density property, we say that M has the density property. If (M,ω) is a calibrated complex manifold, i.e., ω is a nondegenerate holomorphic n-form (n = dimCM) on M , we say that (M,ω) has the volume density property if the Lie algebra XO(M,ω) of all holomorphic vector fields with vanishing ω-divergence has the density property.
118317956
Lectures on Harmonic Maps
A presentation of research on harmonic maps, based on lectures given at the University of California, San Diego. Schoen has worked to use the Fells/Sampson theorem to reprove the rigidity theorem of Masfow and superrigidity theorem of Marqulis. Many of these developments are recorded here.
124274146
On the Density and the Volume Density Property
This article gives a short introduction into the notions of density property (DP) and volume density property (VDP). Moreover we develop an effective criterion of verifying whether a given X has VDP. As an application of this method we give a new proof of the basic fact that the product of two Stein manifolds with VDP admits VDP.
17001843
A pr 2 00 6 HOLOMORPHIC CURVES IN COMPLEX SPACES
We study the existence of closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any non compact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. We also construct a Stein neighborhood basis of any compact complex curve with C 2 boundary in a complex space.
16428953
Strongly pseudoconvex domains as subvarieties of complex manifolds
In this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold $X$ is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on $X$. Examples show that our conditions cannot be weakened in general. We obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with Griffiths positive normal bundle; in the projective case these generalize classical theorems of Remmert, Bishop and Narasimhan concerning proper holomorphic maps and embeddings to ${\Bbb C}^n ={\Bbb P}^n \backslash {\Bbb P}^{n-1}$..
119622686
On the Hodge conjecture for $q$-complete manifolds
We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<\dim X$. This holds in particular for the complement $X=\mathbb{C}\mathbb{P}^n\setminus A$ of any complex projective manifold defined by $q
15928068
DEFORMATIONS OF STEIN STRUCTURES AND EXTENSIONS OF HOLOMORPHIC MAPPINGS
Assume that A is a closed complex subvariety of a Stein manifold X and that f : X ! Y is a continuous map to a complex manifold Y such that the restriction f|A: A ! Y is holomorphic on A. After a homotopic deformation of the Stein structure outside a neighborhood of A in X we find a holomorphic map e f : X ! Y which agrees with f on A and which is homotopic to f relative to A. When dimC X = 2 we must also change the C1 structure on X\A.
12969940
Stein structures and holomorphic mappings
We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions, the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148(2): 619–693, 1998; J Symplectic Geom 3:565–587, 2005).
120085133
Proper Holomorphic Embeddings of Riemann Surfaces with Arbitrary Topology into ℂ2
We prove that given an open Riemann surface $\mathcal{N}$ of arbitrary (finite or infinite) topology, there exists an open domain $\mathcal{M}\subset \mathcal{N}$ homeomorphic to $\mathcal{N}$ which properly holomorphically embeds in ℂ2. Furthermore, $\mathcal{M}$ can be chosen with hyperbolic conformal type.In particular, any open orientable surface M admits a complex structure $\mathcal{C}$ such that $(M,\mathcal{C})$ can be properly holomorphically embedded into ℂ2.
62893679
Embedding some bordered Riemann surfaces int the affine plane
We study the existence of proper holomorphic embeddings of bordered Riemann surfaces into the complex plane C^2. Denote by M(R) the moduli space consisting of all equivalence classes of complex structures J on a given smooth oriented bordered surface R. We introduce a class F(R) in M(R)with the following properties: (1) F(R) is nonempty and open (in a natural topology on M(R)); (2) The interior of any Riemann surface (R,J) in the class F(R) admits a proper holomorphic embedding in C^2; (3) If R is a finitely connected planar domain then F(R)=M(R); (4) Each hyperelliptic bordered Riemann surface (R,J) belongs to the class F(R) and hence admits a proper holomorphic embedding in C^2. Part (3) above is equivalent to the theorem of Globevnik and Stensones (Holomorphic embeddings of planar domains into C^2, Math. Ann. 303, 579-597, 1995). Our approach builds upon the earlier work of Cerne and Globevnik (On holomorphic embedding of planar domains into C^2, J. d'Analyse Math. 8, 269-282, 2000).
119570058
Some aspects of holomorphic mappings: A survey
This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the reflection principle, the scaling method, and the Kobayashi–Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.
129939502
Proper Holomorphic Mappings : A Survey *
A continuous mapping I: X --> Y is called proper if 11(K) is a compact subset of X whenever K is a compact subset of Y. If X and Y are complex spaces and I: X --> Y is a proper holomorphic mapping, then 1-(y) is a compact subvariety of X for all points y E Y. Consequently, if the space X is Stein, the preimage 1(y) is finite for all y E Y. A special class of proper holomorphic maps are the biholomorphic maps, i.e. , the bijective holomorphic maps with a holomorphic inverse. Proper holomorphic mappings between complex spaces were studied in the 1950s and early 1960s; (see Remmert-Stein (RS]). Perhaps the most important result from this time is the Grauert's theorem on the coherence of direct images of a coherent analytic sheaves (Nar2]. A special but important case of this result is the Proper Mapping Theorem of Remmert (GR): If I: X _, Y is a proper holomorphic mapping of complex spaces, and if A C X is a complex subvariety of X, then its image f (A) is a complex subvariety of Y. If both X and Y are Stein spaces and A C X is an irreducible subvariety of dimension k, then B = l(A) is an irreducible subvariety of Y of dimension k. Moreover, there exists a proper, nowhere dense subvari-
31184272
A tranversality theorem for holomorphic mappings and stability of Eisenman-Kobayashi measures
We show that Thom’s Transversality Theorem is valid for holomorphic mappings from Stein manifolds. More precisely, given such a mapping f : S → M from a Stein manifold S to a complex manifold M and given an analytic subset A of the jet space Jk(S,M), f can be approximated in neighborhoods of compacts by holomorphic mappings whose k-jet extensions are transversal to A. As an application the stability of Eisenman-Kobayshi intrinsic k-measures with respect to deleting analytic subsets of codimension > k is proven. This is a generalization of the Campbell-Howard-Ochiai-Ogawa theorem on stability of Kobayashi pseudodistances.
126057839
A relative embedding theorem for Stein spaces.
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120780245
Oka’s principle for holomorphic sections of elliptic bundles
2. The h-principle for locally trivial fibrations 2.1. The h-principle over small subsets 2.2. The h-principle over totally real submanifolds 2.3. Local extension of the h-principle 2.4. Localizable extensions 2.5. The h-principle for Cm-bundles 2.6. Manifolds with totally real souls 2.7. Totally real extensions 2.8. Nicely localizable extensions 2.9. The basic h-principle for Z -* X 2.10. Homomorphisms and holomorphic maps of rank > r
13338106
Survey of Oka theory
Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the first-named author at the conference RAFROT 2010 in Rincon, Puerto Rico, on March 22, 2010, and of a brief survey article by the second-named author in Notices Amer. Math. Soc., January 2010.
9613825
Optimality for totally real immersions and independent mappings of manifolds into C^N
The optimal target dimensions are determined for totally real immersions and for independent mappings into complex affine spaces. Our arguments are similar to those given by Forster, but we use orientable manifolds as far as possible and so are able to obtain improved results for orientable manifolds of even dimension. This leads to new examples showing that the known immersion and submersion dimensions for holomorphic mappings from Stein manifolds to affine spaces are best possible.
119173387
Oka properties of ball complements
Let $$n>1$$n>1 be an integer. We prove that holomorphic maps from Stein manifolds $$X$$X of dimension $${<}n$$<n to the complement $$\mathbb {C}^n{\setminus } L$$Cn\L of a compact convex set $$L\subset \mathbb {C}^n$$L⊂Cn satisfy the basic Oka property with approximation and interpolation. If $$L$$L is polynomially convex then the same holds when $$2\dim X \le n$$2dimX≤n. We also construct proper holomorphic maps, immersions and embeddings $$X\rightarrow \mathbb {C}^n$$X→Cn with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.
118963833
Principles of Algebraic Geometry
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.
16942650
Proper holomorphic embeddings of finitely and some infinitely connected subsets of into
We show that any finitely connected domain can be properly embedded into . For some sequences can also be properly embedded into .
119593373
Embedding Riemann Surfaces Properly into $\CC^2$
For certain bordered submanifolds $M\subset\CC^2$ we show that $M$ can be embedded properly and holomorphically into $\CC^2$. An application is that any subset of a torus with two boundary components can be embedded properly into $\CC^2$.
16711757
Embedding Subsets of Tori Properly into C
Let T be a torus. We prove that all subsets of T with finitely many boundary components (none of them being points) embed properly into C 2. We also show that the algebras of analytic functions on certain countably connected subsets of T are doubly generated.
15848716
Bordered Riemann surfaces in C2
Abstract We prove that the interior of any compact complex curve with smooth boundary in C 2 admits a proper holomorphic embedding into C 2 . In particular, if D is a bordered Riemann surface whose closure admits a holomorphic embedding into C 2 , then D admits a proper holomorphic embedding into C 2 .
119613460
Embeddings of infinitely connected planar domains into C^2
We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many of the punctures belong to the closure of the set of complementary discs.
120706258
Fixed points, Koebe uniformization and circle packings
A domain in the Riemann sphere \(\hat{\mathbb{C}}\) is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following conjecture, known as Koebe’s Kreisnormierungsproblem: Any plane domain is conformally homeomorphic to a circle domain in \(\hat{\mathbb{C}}\). When the domain is simply connected, this is the content of the Riemann mapping theorem.
123094831
Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2
We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C2, admit a proper holomorphic embedding into C2. We also prove that certain infinitely connected open subsets D⊂ℂ admit a proper holomorphic embedding into ℂ2.
119154445
A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗
Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ℂ×ℂ∗, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ℂ×ℂ∗ of recent results of Wold and Forstnerič on the long-standing problem of properly embedding open Riemann surfaces into ℂ2, with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.
117865525
Embeddings of the line in the plane.
In their paper of this title [1], Abhyankar and Moh have shown that (up to algebraic automorphism) the only embedding of the line in the plane, in characteristic 0, is the Standard one. For the complex numbers, this is a strong sort of unknotting theorem. In this paper, using only some reasonably elementary theory of knots, I give a new proof of the theorem of Abhyankar and Moh; this proof is intrinsically topological, and does not appear merely to be a translation of their original proof using "high-school algebra".
117483667
The Geometry of Complex Domains
Preface -- 1 Preliminaries -- 2 Riemann Surfaces and Covering Spaces -- 3 The Bergman Kernel and Metric -- 4 Applications of Bergman Geometry -- 5 Lie Groups Realized as Automorphism Groups -- 6 The Significance of Large Isotropy Groups -- 7 Some Other Invariant Metrics -- 8 Automorphism Groups and Classification of Reinhardt Domains -- 9 The Scaling Method, I -- 10 The Scaling Method, II -- 11 Afterword -- Bibliography -- Index.
189829949
Approximation of biholomorphic mappings by automorphisms of Cn
In this paper we prove several results on approximation of biholomorphic mappings between domains in C" (n _>-2) by holomorphic automorphisms of C". Recall that the group of holomorphic automorphisms of C", denoted Aut C", consists of those holomorphic mappings ~ : C "--, C" that have a holomorphic inverse 4 1 : C" --* C". With the topology of uniform convergence on compact sets, Aut C" is a topological group. While the group Aut C 1 of automorphisms of the complex plane consists only of the linear mappings az + b (a, b e C, a 4= 0), the group Aut C" is very large and complicated when n > 2. Let us choose a linear coordinate system on C" and write the coordinates as z = (z ' ,w), where Z'~-(Z1, . . . ,Zn_I)C--C n 1 and w = z ,~ C. Then AutC n contains mappings of the form
37836447
Subvarieties of ℂn with non-extendable automorphisms
Abstract We investigate algebraic and analytic subvarieties of ℂn with automorphisms which cannot be extended to the ambient space.
119173758
An embedding of {\bf C} in {\bf C}$^2$ with hyperbolic complement
Let $X$ be a closed, $1$-dimensional, complex subvariety of $\CC^2$ and let $\ol{\BB}$ be a closed ball in $\CC^2 - X$. Then there exists a Fatou-Bieberbach domain $\Omega$ with $X \subseteq \Omega \subseteq \CC^2 - \ol{\BB}$ and a biholomorphic map $\Phi: \Omega \ra \CC^2$ such that $\CC^2 - \Phi(X)$ is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in $\CC^2$ whose complement is hyperbolic, and there is a nontrivial Fatou-Bieberbach domain containing any finite collection of complex lines.
119673150
Interpolation by holomorphic automorphisms and embeddings in Cn
Let n > 1 and letCndenote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:Cn →Cnand for holomorphic automorphisms ofCnon discrete subsets ofCn.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds intoCn.For each closed complex submanifold (or subvariety) M ⊂Cnof complex dimension m < n we construct a domain Ω ⊂Cncontaining M and a biholomorphic map F: Ω →CnontoCnwith J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:Cn−m →Cnat infinitely many points. If m = n − 1, we construct F as above such thatCn ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:Cm →Cm−1such that the complementCm+1 ∖F(Cm)is hyperbolic.
120791548
Non-equivalent embeddings into complex euclidean spaces
We study the number of equivalence classes of proper holomorphic embeddings of a Stein space X into ℂn. In this paper we prove that if the automorphism group of X is a Lie group and there exists a proper holomorphic embedding of X into ℂn, 0 < dim X < n, then for any k ≥ 0 there exist uncountably many non-equivalent proper holomorphic embeddings Φ: X × ℂk ↪ ℂn × ℂk. For k = 0 all embeddings will be proved to satisfy the additional property of ℂn\Φ(X) being (n - dim X)-Eisenman hyperbolic. As a corollary we conclude that there are uncountably many non-equivalent proper holomorphic embeddings of ℂk into ℂn whenever 0 < k < n.
15148483
A Fatou-bieberbach Domain in C 2 Which Is Not Runge
Since a paper by J.P.Rosay and W.Rudin from 1988 there has been an open question whether all Fatou-Bieberbach domains are Runge. We give an example of a Fatou-Bieberbach domain Ω in C 2 which is not Runge. The domain Ω provides (yet) a negative answer to the problem of Bremermann.
17427277
Algebraic volume density property of affine algebraic manifolds
We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of ℂn equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.
119688827
Proper holomorphic discs in C^2
In this paper we investigate the global behavior of proper holomorphic maps f from the unit disc U={|z|<1} to C^2. The fact that U is transcendental imposes certain restrictions on the image f(U). For instance, f(U) cannot be contained in any proper complex cone in C^2 since this would force it to be algebraic. On the other hand, we show that a real cone in C^2 with axis R^2 contains the image of a proper holomorphic map f from U to C^2 if and only if the angle of the cone is larger than pi/2. We also construct maps f as above whose images avoid both coordinate axes in C^2. Equivalently, we construct a pair of positive harmonic functions u, v on U such that max{u(z),v(z)} tends to plus infinity when z tends to the boundary of U. Furthermore we show that the components f_1, f_1 of a proper holomorphic map from U to C^2, as well as polynomial and certain rational functions of f_1 and f_2, have the property that their essential range at any boundary point of U omits at most a polar set in C.
18006197
Proper discs in Stein manifolds avoiding complete pluripolar sets
We prove the following theorem: Let X be a Stein manifold of dimension at least 2 and Y a closed complete pluripolar subset of X. Given a point p in the complement of Y there is a proper holomorphic map f from the unit disc to X such that f(0)=p and the image of f avoids Y.
16828626
Proper holomorphic disks in the complement of varieties in \C^2
For any closed analytic set X in C^2 there exists a proper holomorphic embedding of the unit disk into C^2 such that the image avoids X.
15015707
Discs in Stein manifolds
1. The result Let be the open unit disc in C. In the paper we prove the following THEOREM Let M be a Stein manifold, dimM 2. Given a point p 2 M and a vector X tangent to M at p there is a proper holomorphic map f: ! M such that f(0) = p and such that f 0 (0) = X for some > 0. This has been known in the special case when M is a bounded domain in C N with boundary of class C 2. It was proved in FG] by using the fact that for such domains there are bounded strictly plurisubharmonic exhaustion functions without critical points near the boundary. 2. Outline of the proof We describe the idea of the proof in the special case when M is a pseudoconvex domain in C N. Let D C N be a pseudoconvex domain and let be a strictly plurisubharmonic exhaustion function for D. The idea is to start with an analytic disc f : ! D such that f(0) = p; f 0 (0) = X for some > 0, and then to push f(); 2 b in directions that are approximately tangent to the level sets of while keeping f(0) and f 0 (0) xed, keeping f() essentially unchanged for all belonging to a xed compact subset of , and lowering (f()); 2 , only a little. We perform this inductively. If we want to get a proper map in the limit we must, at each step, increase (f()); 2 b, for a certain amount and the sum of these amounts must equal +1. However, this amount depends on the lower bound of jgradj. In particular , if a < b and if has no critical value on a; b] then there is a > 0 such that whenever a (f()) b (2 b), then, in the pushing process one can increase (f()) for at least for each 2 b. Thus, if has no critical values on (b; 1) one can go on and use the pushing process inductively to obtain a proper holomorphic map f: ! D such that f(0) = p. In this way one can prove the theorem in the special case when M is a bounded pseudoconvex domain in C N with boundary of class C 2 FG]. In the convex setting the idea of pushing boundaries of …
122702857
Immersions and embeddings in domains of holomorphy
Let Dx be a bounded smooth strongly pseudoconvex domain in C^ and let D2 be a domain of holomorphy in CM (2 < N, 5 < M, IN < M). There exists then a proper holomorphic immersion from Dx to D2 . Furthermore if PI(DX, D2) is the set of proper holomorphic immersions from Dx to D2 and A(DX,D2) is the set of holomorphic maps from Dx to D2 that are continuous on the boundary, then the closure of Pl(Dx , D2) in the topology of uniform convergence on compacta contains A(DX, D2). The approximating proper maps can be made tangent to any finite order of contact at a given point. The same result was obtained for proper holomorphic maps, in one codimension, when the target domain has a plurisubharmonic exhaustion function with no saddle critical points. This includes the case where the target domain is convex. Density in a weaker sense was derived in one codimension when the critical points are contained in a compact subset of the target domain. This occurs (for example) when the target domain is bounded weakly pseudoconvex with C2-smooth boundary. If the target domain is strongly pseudoconvex then the approximating proper holomorphic maps can also be made continuous on the boundary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in CL is called "Mdimensional-pseudoconvex" (where L > M) if it has a smooth exhaustion function r such that every point w in this domain has some M-dimensional complex affine subspace going through this point for which r, restricted to this subspace, is strictly plurisubharmonic in w . In the result mentioned above the assumption that the target domain is pseudoconvex in CM (M > 2N, 5) can be substituted for the assumption that the domain is "M-dimensionalpseudoconvex". Similarly, the assumption that the target domain D2 is "(N + 1 )-dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are "(N+1 )-dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from Dx to D2 contains A(DX, D2). All the results are obtained with embeddings when the Euclidean dimensions are such that dimc(£>2) > 2dimc(öi) + 1 . Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from D\ to D2 contains A{DX, D2).
10691259
Approximation of holomorphic mappings on strongly pseudoconvex domains
Abstract Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold S. We prove that for every complex manifold Y the set 𝒜(D, Y) of all continuous maps → Y which are holomorphic in D is a complex Banach manifold, and every ƒ ∈ 𝒜(D, Y) can be approximated uniformly on by maps holomorphic in open neighborhoods of in S. Analogous results are obtained for maps of class 𝒜 r (D), r ∈ . We also establish the Oka property for sections of continuous or smooth fiber bundles over which are holomorphic over D and whose fiber enjoys the Convex Approximation Property (Theorem 1.7).
121755148
A Complete Bounded Complex Submanifold of C 3
We produce an example of a bounded complete complex submanifold of C3. This is accomplished by using the duality between H '(I) and BMO(T). The question of whether there exists a complete bounded complex submanifold of C' has been an open problem (see [3] for definitions and a discussion of this problem). We present here a method of producing such submanifolds. Suppose that f1(z) and f2(z) are two functions which are analytic and bounded in the unit disk A of C, and suppose that these two functions have the property f{ Ifl(z)I + If'(z)I da(z) = oo(1.1) for all curves F c A which terminate on aA = T. (Here a denotes Euclidean arc length.) Then z e A -) (z, f1(z), f2(z)) is an embedded complete bounded complex submanifold of C3. To construct two bounded analytic functions f1 and f2 satisfying (1.1) we use C. Fefferman's theorem [1] that every real valued function g)p E BMO(T) can be represented by gp = u + 1v, u, v E L'(T). Here 1v denotes the Hilbert transform of v. Consider the harmonic function 00 10" qp(rei') = E -cos lOW. Then IV9gp(z)I > lOn/lOOn if z is in the annulus An = {z: 1 11 * 10-n-I < IzI < 1 -9 10-"-'n . To see this, note that IV((l/n)rl0" cos 10')I is of order of magnitude lO"/n on An. The term lVnz ''? cos I 10'6) is small on An because it is bounded pointwise by 21j27 10'/j. The term
33184269
Every bordered Riemann surface is a complete proper curve in a ball
We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of $$\mathbb C ^2$$, and a complete proper holomorphic embedding into a ball of $$\mathbb C ^3$$.
32285439
Complete bounded embedded complex curves in C^2
We prove that any convex domain of C^2 carries properly embedded complete complex curves. In particular, we exhibit the first examples of complete bounded embedded complex curves in C^2
119659561
Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls
We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.
117868974
A complete complex hypersurface in the ball of C^N
In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : M^k --> C^N with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. The principal result of the present paper is a construction of a holomorphic function on the open unit ball B_N of C^N whose real part is unbounded on every path in B_N of finite length that ends on the boundary of B_N. A consequence is the existence of a complete, closed, complex hypersurface in B_N. This gives a positive answer to Yang's question in all dimensions k, N, 1\leq k
119643597
Embedding complete holomorphic discs through discrete sets
Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for any path p:[0,1)-->U such that |p(t)|-->1 as t-->1, the path t--> f(p(t)) from [0,1) to B has infinite length.
117134897
Holomorphic functions unbounded on curves of finite length
Given a pseudoconvex domain , we prove that there is a holomorphic function f on D such that the lengths of paths $$p:\ [0,1] \rightarrow D$$p:[0,1]→D along which $$\mathfrak {R}f$$Rf is bounded above, with p(0) fixed, grow arbitrarily fast as $$p(1)\rightarrow bD$$p(1)→bD. A consequence is the existence of a complete closed complex hypersurface $$ M\subset D$$M⊂D such that the lengths of paths $$p:\ [0,1]\rightarrow M$$p:[0,1]→M, with p(0) fixed, grow arbitrarily fast as $$p(1)\rightarrow bD$$p(1)→bD.
119690913
A construction of complete complex hypersurfaces in the ball with control on the topology
<jats:p>Given a
closed complex hypersurface <jats:inline-formula id="j_crelle-2016-0061_ineq_9999_w2aab3b7e5890b1b6b1aab1c14b1b1Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0213.png" /><jats:tex-math>{Z\subset\mathbb{C}^{N+1}}</jats:tex-math></jats:alternatives></jats:inline-formula> (<jats:inline-formula id="j_crelle-2016-0061_ineq_9998_w2aab3b7e5890b1b6b1aab1c14b1b3Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0190.png" /><jats:tex-math>{N\in\mathbb{N}}</jats:tex-math></jats:alternatives></jats:inline-formula>) and a compact subset <jats:inline-formula id="j_crelle-2016-0061_ineq_9997_w2aab3b7e5890b1b6b1aab1c14b1b5Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0179.png" /><jats:tex-math>{K\subset Z}</jats:tex-math></jats:alternatives></jats:inline-formula>, we prove the existence of a pseudoconvex Runge domain <jats:italic>D</jats:italic> in <jats:italic>Z</jats:italic> such that <jats:inline-formula id="j_crelle-2016-0061_ineq_9996_w2aab3b7e5890b1b6b1aab1c14b1c11Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0177.png" /><jats:tex-math>{K\subset D}</jats:tex-math></jats:alternatives></jats:inline-formula> and there is a complete proper holomorphic embedding from <jats:italic>D</jats:italic> into the unit ball of <jats:inline-formula id="j_crelle-2016-0061_ineq_9995_w2aab3b7e5890b1b6b1aab1c14b1c15Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0342.png" /><jats:tex-math>{\mathbb{C}^{N+1}}</jats:tex-math></jats:alternatives></jats:inline-formula>.
For <jats:inline-formula id="j_crelle-2016-0061_ineq_9994_w2aab3b7e5890b1b6b1aab1c14b1c17Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0188.png" /><jats:tex-math>{N=1}</jats:tex-math></jats:alternatives></jats:inline-formula>, we derive the existence of complete properly embedded complex curves in the unit ball of <jats:inline-formula id="j_crelle-2016-0061_ineq_9993_w2aab3b7e5890b1b6b1aab1c14b1c19Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0340.png" /><jats:tex-math>{\mathbb{C}^{2}}</jats:tex-math></jats:alternatives></jats:inline-formula>, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc <jats:inline-formula id="j_crelle-2016-0061_ineq_9992_w2aab3b7e5890b1b6b1aab1c14b1c21Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0350.png" /><jats:tex-math>{\mathbb{D}\subset\mathbb{C}}</jats:tex-math></jats:alternatives></jats:inline-formula> into the unit ball of <jats:inline-formula id="j_crelle-2016-0061_ineq_9991_w2aab3b7e5890b1b6b1aab1c14b1c23Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0340.png" /><jats:tex-math>{\mathbb{C}^{2}}</jats:tex-math></jats:alternatives></jats:inline-formula>.
These are the first known examples of complete bounded embedded complex hypersurfaces in <jats:inline-formula id="j_crelle-2016-0061_ineq_9990_w2aab3b7e5890b1b6b1aab1c14b1c25Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0061_eq_0342.png" /><jats:tex-math>{\mathbb{C}^{N+1}}</jats:tex-math></jats:alternatives></jats:inline-formula> with any control on the topology.</jats:p>
119319002
Minimal surfaces in minimally convex domains
In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb{R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb{R}^n$ for any $n\ge 3$ by sequences of conformal minimal disks whose boundaries converge to $K$ in the measure theoretic sense.
40808295
The Calabi–Yau problem, null curves, and Bryant surfaces
In this paper we prove that every bordered Riemann surface $$M$$M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean 3-space $$\mathbb {C}^3$$C3. The real part of such an embedding is a complete conformal minimal immersion $$M\rightarrow \mathbb {R}^3$$M→R3 with bounded image. For any such $$M$$M we also construct proper null holomorphic embeddings $$M\hookrightarrow \mathbb {C}^3$$M↪C3 with a bounded coordinate function; these give rise to properly embedded null curves $$M\hookrightarrow SL_2(\mathbb {C})$$M↪SL2(C) and to properly immersed Bryant surfaces $$M\rightarrow \mathbb {H}^3$$M→H3 in the hyperbolic 3-space. In particular, we provide the first examples of proper Bryant surfaces with finite topology and of hyperbolic conformal type. The main novelty when compared to the existing results in the literature is that we work with a fixed conformal structure on $$M$$M. This is accomplished by introducing a conceptually new method based on complex analytic techniques. One of our main tools is an approximate solution to certain Riemann-Hilbert boundary value problems for null curves in $$\mathbb {C}^3$$C3, developed in Sect. 3.
11039894
Noncritical holomorphic functions on Stein manifolds
We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).
121744180
Boundary Behaviour of Conformal Maps
1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve Families and Capacity.- 10. Hausdorff Measure.- 11. Local Boundary Behaviour.- References.- Author Index.
119602018
A properly embedded holomorphic disc in the ball with finite area and dense boundary
In this paper we construct a properly embedded holomorphic disc in the unit ball $$\mathbb {B}^2$$B2 of $$\mathbb {C}^2$$C2 having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $$\mathbb {B}^2$$B2; on the other hand, its boundary curve is everywhere dense in the sphere $$b\mathbb {B}^2$$bB2. A similar result is proved in higher dimensions. Our construction is based on an approximation result in contact geometry, also proved in the paper.
119305328
The generalized Oka-Grauert principle for 1-convex manifolds
This paper presents a proof of the generalized Oka-Grauert principle for 1-convex manifolds: Every continuous mapping from a 1-convex manifold X to a complex manifold Y which is already holomorphic on a neighborhood of the exceptional set is homotopic to a holomorphic one provided that either Y satisfies CAP or we are free to change the complex structure on X.
122502085
TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2
In this paper I give a completed topological characterization of Stein manifolds of complex dimension >2. Another paper (see [E14]) is devoted to new topogical obstructions for the existence of a Stein complex structure on real manifolds of dimension 4. Main results of the paper have been announced in [E13].
118671586
From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
A beautiful and comprehensive introduction to this important field. --Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a super introduction to this area and also contains the authors' new results. --Tomasz Mrowka, MIT This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from "Stein to Weinstein") and its applications in the complex geometric world of Stein manifolds (the road "back"). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology. Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, $h$-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.|This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine complex (a.k.a. Stein) manifolds have canonically built into them symplectic geometry which is responsible for many phenomena in complex geometry and analysis. The goal of the book is the exploration of this symplectic geometry (the road from ""Stein to Weinstein"") and its applications in the complex geometric world of Stein manifolds (the road ""back""). This is the first book which systematically explores this connection, thus providing a new approach to the classical subject of Stein manifolds. It also contains the first detailed investigation of Weinstein manifolds, the symplectic counterparts of Stein manifolds, which play an important role in symplectic and contact topology. Assuming only a general background from differential topology, the book provides introductions to the various techniques from the theory of functions of several complex variables, symplectic geometry, $h$-principles, and Morse theory that enter the proofs of the main results. The main results of the book are original results of the authors, and several of these results appear here for the first time. The book will be beneficial for all students and mathematicians interested in geometric aspects of complex analysis, symplectic and contact topology, and the interconnections between these subjects.