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.

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title: Oka’s principle for holomorphic sections of elliptic bundles
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title: Embedding Riemann Surfaces Properly into $\CC^2$
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title: Embeddings of infinitely connected planar domains into C^2
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title: Fixed points, Koebe uniformization and circle packings
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title: Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2
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title: A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗
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title: Embeddings of the line in the plane.
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title: The Geometry of Complex Domains
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title: Approximation of biholomorphic mappings by automorphisms of Cn
abstract: 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

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title: An embedding of {\bf C} in {\bf C}$^2$ with hyperbolic complement
abstract: 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.

title: Interpolation by holomorphic automorphisms and embeddings in Cn
abstract: 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.

title: Non-equivalent embeddings into complex euclidean spaces
abstract: 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.

title: A Fatou-bieberbach Domain in C 2 Which Is Not Runge
abstract: 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.

title: Algebraic volume density property of affine algebraic manifolds
abstract: 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.

title: Proper holomorphic discs in C^2
abstract: 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.

title: Proper discs in Stein manifolds avoiding complete pluripolar sets
abstract: 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.

title: Proper holomorphic disks in the complement of varieties in \C^2
abstract: 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.

title: Discs in Stein manifolds
abstract: 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 …

title: Immersions and embeddings in domains of holomorphy
abstract: 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).

title: 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).

title: A Complete Bounded Complex Submanifold of C 3
abstract: 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  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  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

title: Every bordered Riemann surface is a complete proper curve in a ball
abstract: 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 $$\mathbb C ^3$$.

title: Complete bounded embedded complex curves in C^2
abstract: 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

title: Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls
abstract: 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.

title: A complete complex hypersurface in the ball of C^N
abstract: 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

title: Embedding complete holomorphic discs through discrete sets
abstract: 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.

title: Holomorphic functions unbounded on curves of finite length
abstract: 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.

title: A construction of complete complex hypersurfaces in the ball with control on the topology
abstract: <jats:p>Given a
closed complex hypersurface <jats:inline-formula id="j_crelle-2016-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-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-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>

title: Minimal surfaces in minimally convex domains
abstract: 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.

title: The Calabi–Yau problem, null curves, and Bryant surfaces
abstract: 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.

title: Noncritical holomorphic functions on Stein manifolds
abstract: 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).

title: Boundary Behaviour of Conformal Maps
abstract: 1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4.  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.

title: A properly embedded holomorphic disc in the ball with finite area and dense boundary
abstract: 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.

title: The generalized Oka-Grauert principle for 1-convex manifolds
abstract: 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.

title: TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2
abstract: 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].

title: From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
abstract: 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.