Symplectic Geometry of Stein Manifolds K. Cieliebak and Y. Eliashberg Incomplete draft, October 12, 2007 2 Contents 1 J-convex functions and hypersurfaces 5 1.1 Linear algebra . 5 1.2 J-convex functions . 6 1.3 The Levi form of a hypersurface . 7 1.4 J-convexity and geometric convexity . 11 1.5 Examples of J-convex ::: ...................... 14 1.6 J-convex ::: in Cn .......................... 16 2 Smoothing 21 2.1 J-convexity and plurisubharmonicity . 21 2.2 Smoothing of J-convex functions . 24 2.3 Critical points of J-convex functions . 29 2.4 From families of hypersurfaces to J-convex functions . 34 3 Symplectic and Contact Preliminaries 37 3.1 Symplectic vector spaces . 37 3.2 Symplectic vector bundles . 39 3.3 Symplectic manifolds . 40 3.4 Moser's trick and symplectic normal forms . 42 3.5 Contact manifolds . 44 3.6 Stabilization of Legendrian submanifolds . 50 4 The h-principles 53 4.1 Immersions and embeddings . 53 4.2 The h-principle for isotropic immersions . 56 4.3 The h-principle for isotropic embeddings . 57 3 4 CONTENTS 4.4 The h-principle for totally real embeddings . 58 4.5 Discs attached to J-convex boundary . 58 4.6 The three-dimensional case . 60 5 Weinstein structures 61 5.1 Convex symplectic manifolds . 61 5.2 Deformations of convex sympletic structures . 62 5.3 Weinstein manifolds . 63 5.4 Weinstein structure of a Stein manifold . 66 6 Weinstein handlebodies 69 6.1 Handles in the smooth category . 69 6.2 Weinstein handlebodies . 72 6.2.1 Standard handle . 72 6.2.2 Attaching a handle to a Weinstein domain . 73 6.3 Subcritical Weinstein manifolds . 77 6.4 Morse-Smale theory for Weinstein structures . 79 7 Shapes for J-convex hypersurfaces 81 7.1 J-convex surroundings and extensions . 81 7.2 Shapes for J-convex hypersurfaces . 84 7.3 Construction of special shapes . 88 7.4 Families of special shapes . 89 7.5 J-convex surroundings and extensions on a handle . 96 7.6 Proof of the surrounding and extension theorems . 98 8 Real analytic approximations 101 8.1 Some complex analysis on Stein manifolds . 101 8.2 Real analytic approximations . 105 9 Extension of Stein structures over handles 113 9.1 Handles in the holomorphic category . 113 9.2 Extension of Stein structures over handles . 115 Chapter 1 J-convex functions and hypersurfaces 1.1 Linear algebra A complex vector space (V; J) is a real vector space V of dimension 2n with an endomorphism J satisfying J 2 = −1l. A Hermitian form on (V; J) is an R- bilinear map H : V × V ! C which is C-linear in the first variable and satisfies H(X; Y ) = H(Y; X). If H is, moreover, positive definite it is called Hermitian metric. We can write a Hermitian form H uniquely as H = g − i!; where g is a symmetric and ! a skew-symmetric bilinear form on the real vector space V . The forms g and ! determine each other: g(X; Y ) = !(X; JY );!(X; Y ) = g(JX; Y ) for X; Y 2 V . Moreover, the forms ! and g are invariant under J, which can be equivalently expressed by the equation !(JX; Y ) + !(X; JY ) = 0: Conversely, given a skew-symmetric J-invariant form !, we can uniquely recon- struct the corresponding Hermitian form H: H(X; Y ) := !(X; JY ) − i!(X; Y ): (1.1) n For example, consider the complex vector space (C ; i) with coordinates z1 = x1 + iy1; : : : ; zn = xn + iyn. It carries the standard Hermitian metric n X (v; w) := vjw¯j = hv; wi − i!0(v; w); j=1 5 6 CHAPTER 1. J-CONVEX FUNCTIONS AND HYPERSURFACES P where h·; ·i is the Euclidean metric and !0 = j dxj ^ dyj the standard sym- plectic form on Cn. 1.2 J-convex functions An almost complex structure on a smooth manifold V of real dimension 2n is an automorphism J : TV ! TV satisfying J 2 = −1l on each fiber. The pair (V; J) is called almost complex manifold. It is called complex manifold if J is integrable, i.e. J is induced by complex coordinates on V . By the theorem of Newlander and Nirenberg [38], a (sufficiently smooth) almost complex structure J is integrable if and only if its Nijenhuis tensor N(X; Y ) := [JX; JY ] − [X; Y ] − J[X; JY ] − J[JX; Y ]; X; Y 2 T V; vanishes identically. In the following let (V; J) be an almost complex manifold. To a smooth function φ : V ! R we associate the 2-form !φ := −ddCφ, where the differential operator dC is defined by dCφ(X) := dφ(JX) for X 2 TV . The form !φ is in general not J-invariant. However, it is J- invariant if J is integrable. To see this, consider the complex vector space (Cn; i). Given a function φ : Cn ! R, define the complex valued (1; 1)-form n X @2φ @@φ¯ := dz ^ dz¯ : @z @z¯ i j i;j=1 i j Using the identities dzj ◦ i = i dzj; dz¯j ◦ i = −i dz¯j we compute X @φ @φ X @φ @φ dCφ = dz ◦ i + dz¯ ◦ i = i dz − i dz¯ ; @z j @z¯ j @z j @z¯ j j j j j j j X @2φ ddCφ = −2i dz ^ dz¯ : @z @z¯ i j i;j i j Hence ¯ !φ = 2i@@φ (1.2) ¯ and the i-invariance of !φ follows from the invariance of @@φ. 1.3. THE LEVI FORM OF A HYPERSURFACE 7 1 A function φ is called J-convex if !φ(X; JX) > 0 for all nonzero tangent vectors X. If !φ is J-invariant it defines by (1.1) a unique Hermitian form Hφ := gφ − i!φ; and φ is J-convex iff the Hermitian form Hφ is positive definite. From (1.2) we can derive a simple expression for the form Hφ associated to a n @2φ n function φ : ! in terms of the matrix aij := . For v; w 2 we C R @zi@z¯j C have X X !φ(v; w) = 2i aijdzi ^ dz¯j(v; w) = 2i aij(viw¯j − wiv¯j) ij ij 0 1 X X = 2i (aijviw¯j − a¯ijv¯iwj) = −4Im @ aijviw¯jA ; ij ij hence n X @2φ H (v; w) = 4 v w¯ : (1.3) φ @z @z¯ i j i;j=1 i j P 2 n Example 1.1. The function φ(z) := j jzjj on C is i-convex with respect to the standard complex structure i. The corresponding form Hφ equals 4( ; ), where ( ; ) is the standard Hermitian metric on Cn. 1.3 The Levi form of a hypersurface Let Σ be a smooth (real) hypersurface in an almost complex manifold (V; J). Each tangent space TpΣ ⊂ TpV , p 2 Σ, contains a unique maximal complex subspace ξp ⊂ TpΣ which is given by ξp = TpΣ \ JTpΣ: Suppose that Σ is cooriented by a transverse vector field ν to Σ in V such that Jν is tangent to Σ. The hyperplane field ξ can be defined by a Pfaffian equation fα = 0g, where the sign of the 1-form α is fixed by the condition α(Jν) > 0. The 2-form !Σ := dαjξ is then defined uniquely up to multiplication by a positive function. As in the previous section we may ask whether !Σ is J-invariant. The following lemma gives a necessary and sufficient condition in terms of the Nijenhuis tensor. 1Throughout this book, by convexity and J-convexity we will always mean strict convexity and J-convexity. Non-strict (J-)convexity will be referred to as weak (J-)convexity. 8 CHAPTER 1. J-CONVEX FUNCTIONS AND HYPERSURFACES Lemma 1.2. The form !Σ is J-invariant for a hypersurface Σ if and only if Njξ×ξ takes values in ξ. The form !Σ is J-invariant for every hypersurface Σ if and only if for all X; Y 2 TV , N(X; Y ) lies in the complex plane spanned by X and Y . In particular, this is the case if J is integrable or if V has complex dimension 2. Proof. Let Σ ⊂ V be a hypersurface and α a defining 1-form for ξ. Extend α to a neighborhood of Σ such that α(ν) = 0. For X; Y 2 ξ we have [X; Y ] 2 T Σ and therefore J[X; Y ] = aν + Z for some a 2 R and Z 2 ξ. This shows that α(J[X; Y ]) = 0 for all X; Y 2 ξ. Applying this to various combinations of X, Y , JX and JY we obtain α N(X; Y ) = α([JX; JY ]) − α([X; Y ]); α JN(X; Y ) = α([X; JY ]) + α([JX; Y ]): The form !Σ is given by 1 1 ! (X; Y ) = X · α(Y ) − Y · α(X) − α([X; Y ] = − α([X; Y ]): Σ 2 2 Inserting this in the formulae above yields 1 − α N(X; Y ) = ! (JX; JY ) − ! (X; Y ); 2 Σ Σ 1 − α JN(X; Y ) = ! (X; JY ) + ! (JX; Y ): 2 Σ Σ Hence the J-invariance of !Σ is equivalent to α N(X; Y ) = α JN(X; Y ) = 0; i.e. N(X; Y ) 2 ξ for all X; Y 2 ξ. This proves the first statement and the 'if' in the second statement. For the 'only if' it suffices to note that if N(X; Y ) does not lie in the complex plane spanned by X and Y for some X; Y 2 TV , then we find a hypersurface Σ such that X; Y 2 ξ and N(X; Y ) 2= ξ.