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STEIN STRUCTURES AND HOLOMORPHIC MAPPINGS 7 FRANCFORSTNERICˇ & MARKOSLAPAR 0 0 2 Abstract. WeprovethateverycontinuousmapfromaSteinmanifold n X to a complex manifold Y can be made holomorphic by a homotopic a deformation of both the map and the Stein structure on X. In the ab- J senceoftopologicalobstructionstheholomorphicmapmaybechosento 9 have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact ] V family. Ourmainresultsareactuallyprovedforsmoothalmostcomplex sourcemanifolds(X,J)withthecorrecthandlebodystructure. Thepa- C per contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) . h homotopy characterization of Stein manifolds and a slightly different t explanation of the construction of exotic Stein surfaces due to Gompf a (AnnMath148(2):619–693,1998;JSymplecticGeom3:565–587,2005). m [ 4 v 1. Introduction 2 1 AStein manifoldisacomplexmanifoldwhichisbiholomorphictoaclosed 2 7 complex submanifold of a Euclidean space CN [40]. The following is a sim- 0 plified version of our main results, Theorems 6.1, 6.2 and 7.1. 5 0 Theorem 1.1. Let X be a Stein manifold with the complex structure oper- / h ator J, and let f: X → Y be a continuous map to a complex manifold Y. t a (i) If dimCX 6= 2, there exist a Stein complex structure J on X, homo- m topic to J, and a J-holomorphic map f: X → Y homeotopic to f. : v (ii) If dimCX = 2, theere are an orientationepreserving homeomorphism i h: X → X′ ontoaSteinsurfaceX′ andaholomorphic mapf′: X′ → X Y such that the map f := f′◦h: X → Y is homotopic to f. r a e TheSteinstructure J in(i), andthe homeomorphism hin(ii), canbe chosen the same for all memebers of a compact Hausdorff family of maps X → Y. More precisely, in case (i) we find a smooth homotopy Jt ∈ EndRTX (J2 = −Id, t ∈ [0,1]) consisting of integrable (but not necessarily Stein) t complex structures on the underlying smooth manifold X, connecting the Date: July 5, 2006. 2000MathematicsSubjectClassification. 32H02,32Q30,32Q55,32Q60,32T15,57R17. Key words and phrases. Stein manifolds, complex structures, holomorphic mappings. Supported bygrants P1-0291 and J1-6173, Republicof Slovenia. The original publication is available at http://www.springerlink.com. http://dx.doi.org/10.1007/s00209-006-0093-0. 1 2 FRANCFORSTNERICˇ &MARKOSLAPAR Stein structure J = J with a new Stein structure J = J, such that f 0 1 is homotopic to a J-holomorphic map. In case (ii) we geteessentially the same statement afteer changing the smooth structure on X, i.e., the new Stein structure J on X may be exotic. More precise statements are given by Theorem 6.1 efor part (i), and by Theorem 7.1 for part (ii). The question whether every continuous map from a Stein manifold to a given complex manifold Y is homotopic to a holomorphic map is the central theme of the Oka-Grauert theory. Classical results of Oka [49], Grauert [30, 31, 32] and Gromov [37] give an affirmative answer when Y is a complex homogeneous manifold or, more generally, if it admits a dominating spray. (See also [23] and [21].) Recently this Oka property of Y has been characterized in terms of a Runge approximation property for entire maps Cn → Y on certain special compact convex subset of Cn [19, 20]. The Oka property holds only rarely as it implies in particular that Y is dominated by a complex Euclidean space, and this fails for any compact complex manifold of Kodaira general type. For a discussion of this subject see [21]. Although one cannot always finda holomorphic representative in each homotopy class of maps X → Y, Theorem 1.1 gives a representative which is holomorphic with respect to some Stein structure on X homotopic to the original one. Even if the source complex manifold X is not Stein, we can obtain a holomorphic map in a given homotopy class on a suitable Stein domain in X, provided that X has a correct handlebody structure. Theorem 1.2. Let X be an n-dimensional complex manifold which admits a Morse exhaustion function ρ: X → R without critical points of index > n. Let f: X → Y be continuous map to a complex manifold Y. (i) If n 6= 2, there exist an open Stein domain Ω in X, a diffeomorphism h: X → h(X) = Ω which is diffeotopic to the identity map id on X X, and a holomorphic map f′: Ω → Y such that f′ ◦h: X → Y is homotopic to f. (ii) If n = 2, the conclusion in (i) still holds if ρ has no critical points of index > 1; in the presence of critical points of index 2 the conclusion holds with h a homeomorphism which is homeotopic to id . X Theorem 1.2 immediately implies Theorem 1.1: If h : X → h (X) ⊂ X t t (t ∈ [0,1]) is a diffeotopy from h = id to h = h: X → Ω as in Theorem 0 X 1 1.2 then J := h∗(J) is a homotopy of integrable complex structures on X t t connecting the original structure J = J to a Stein structure J = h∗(J| ), 0 TΩ and f = f′◦h: X → Y is a J-holomorphic map homotopic toef. Ouer proof of Theorem 1.2eshows that the only essential obstruction in finding a holomorphic map in a given homotopy class is that X may be holomorphically ‘too large’ to fit into Y. This vague notion of ‘holomorphic rigidity’ has several concrete manifestations, for example, the distance decreasing property of holomorphic maps in most of the standard biholo- morphically invariant metrics (such as Kobayashi’s). The problem can be STEIN STRUCTURES AND HOLOMORPHIC MAPPINGS 3 avoidedbyrestrictingthesizeofthedomainwhileatthesametimeretaining the topological (and smooth in dimension 6= 2) characteristics of X. The following simple example illustrates that Theorems 1.1 and 1.2 are optimal even for maps of Riemann surfaces. Example 1.3. Let X = A = {z ∈ C: 1/r < |z| < r}, and let Y = A for r R another R > 1. We have [X,Y] = Z. A homotopy class represented by an integer k ∈ Z admits a holomorphic representative if and only if r|k| ≤ R, and in this case a representative is z → zk. Since every complex structure on an annulus is biholomorphic to A for some r > 1, we see that at most r finitely many homotopy classes of maps between any pair of annuli contain a holomorphic map. The conclusion of Theorem 1.1 can be obtained by a radial dilation, decreasing the value of r > 1 to another value satisfying rk ≤ R, which amounts to a homotopic change of the complex structure on X. This allows us to simultaneously deform any compact family of maps X → Y to a family of holomorphic maps, but it is impossible to do it for a sequence of maps belonging to infinitely many different homotopy classes. The problem disappears in the limit as R = +∞ when Y is the complex Lie group C∗ = C\{0} and the Oka-Grauert principle applies [31], [49]. The same phenomenon appears whenever the fundamental group π (Y) contains 1 anelement[α]ofinfiniteordersuchthattheminimalKobayashilengthl of N loopsinY representingtheclassN[α] ∈π (Y)tendsto+∞asN → +∞: A 1 homotopically nontrivial loop γ in X with positive Kobayashi length K (γ) X can be mapped to the class N[α] by a holomorphic map X → Y only if l ≤ K (γ), and this is possible for at most finitely many N ∈ N. N X Our construction also gives holomorphic maps of maximal rank (immer- sionsresp.submersions)providedthattherearenotopological obstructions. The following is a simplified version of Theorem 6.2 below. Theorem 1.4. Let X be a Stein manifold of dimension dimX 6= 2. As- sume that f: X → Y is a continuous map to a complex manifold Y which is covered by a complex vector bundle map ι: TX → f∗(TY) of fiberwise maximal rank. Then there is a Stein structure J on X, homotopic to J, and a J-holomorphic map f: X → Y of pointwisee maximal rank which is homotoepic to f. The analogoeus conclusion holds if dimX = 2 and X admits a Morse exhaustion function ρ: X → R without critical points of index > 1. Theorem 1.4 is a holomorphic analogue of the Smale-Hirsch h-principle for smooth immersions [54, 39, 36] and of the Gromov-Phillips h-principle for smooth submersions [34, 51]. The conclusion holds with a fixed Stein structureon X provided that Y satisfies a certain flexibility condition introduced (for submersions) in [18]. For maps to Euclidean spaces see also [36, §2.1.5] (for immersions) and [17] (for submersions). An important source of Stein manifolds are the holomorphically complete Riemann domains π: X → Cn, π beinga locally biholomorphic map. These 4 FRANCFORSTNERICˇ &MARKOSLAPAR arise as the envelopes of holomorphy of domains in, or over, Cn. Clearly every such manifold is holomorphically parallelizable, but the converse has been a long standing open problem: Does everyn-dimensional Steinmanifold X with atrivial complex tangent bundle admit a locally biholomorphic map π: X → Cn ? In 1967 Gunning and Narasimhan gave a positive answer for open Rie- mann surfaces [38]. In 2003 the first author of this paper proved that every parallelizable Stein manifold Xn admits a holomorphic submersion f: X → Cn−1 [17]; the remaining problem is to find a holomorphic function g onX whoserestrictiontoeach levelsetoff hasnocritical points(themap (f,g): X → Cn is then locally biholomorphic). Theorem 1.4 with Y = Cn (n = dimX) shows that the above problem is solvable up to homotopy: Corollary 1.5. If (X,J) is a Stein manifold of dimension n 6= 2 whose holomorphic tangent bundle TX is trivial then there are a Stein structure J on X, homotopic to J, and a J-holomorphic immersion π: X → Cn. e e NotethateveryclosedcomplexsubmanifoldX ⊂ CN withtrivialcomplex normal bundle TCN| /TX is parallelizable [13]; this holds in particular for X any smooth complex hypersurface in CN. All our main results are actually proved in the class of smooth almost complex manifolds(X,J) whichadmitaMorseexhaustion functionρ: X → R without critical points of index > n = 1 dimRX (see Theorems 6.1, 2 6.2 and 7.1). By Morse theory such X is homotopically equivalent to a CW complex of dimension at most n [46]. Since the Morse indices of any strongly plurisubharmonic exhaustion function satisfy this index condition, this holds for every Stein manifold (Lefshetz [44], Andreotti and Fraenkel [1], Milnor [46]). Conversely, if (X,J) satisfies the above index condition and dimRX 6= 4 then J is homotopic to an integrable Stein structure on X according to Eliashberg [7]. The present paper contains another proof of this important result, with an additional argument provided in the critical case when attaching handles of maximal real dimension n = 1 dimRX. A 2 detailed understanding of this construction is unavoidable for our purposes, and assuming that the initial structure on X is already integrable Stein (as in Theorem 1.1) does not really simplify our proof. The story is even more interesting when dimRX = 4: A smooth oriented four manifold without handles of index > 2 is homeomorphic to a Stein surface, butthe underlying smooth structure must be changed ingeneral(Gompf [27], [28]). Indeed, a closed orientable real surface S smoothly embedded in a Stein surface X (or in a compact Ka¨hler surface with b+(X) > 1), with the only exception of a null-homologous 2-sphere, satisfies the generalized adjunction inequality: (1.1) [S]2+|c (X)·S| ≤ −χ(S). 1 STEIN STRUCTURES AND HOLOMORPHIC MAPPINGS 5 (See Chapter 11 in [29] and the papers [16, 42, 45, 47, 50].) For a 2-sphere the above inequality yields [S]2 ≤ −2. Taking X = S2 × R2 = CP1 ×C, the embedded 2-sphere S2×{0} ⊂ X generates H (X,Z) = Z and satisfies 2 [S]2 = 0, hence X does not admit any non-exotic Stein structure. Nevertheless, there is a bounded Stein domain in C2 homeomorphic to S2×R2. This is a special case of Gompf’s result that for every tamely topo- logically embedded CW 2-complex M in a complex surface X there exists a topological isotopy of X which is uniformly close to the identity on X and which carries M onto a complex M′ ⊂ X with a Stein thickening, i.e., an open Stein domain Ω ⊂ X homeomorphic to the interior of a handlebody with core M [28, Theorem 2.4]. In his proof, Gompf uses kinky handles of index 2 in each place where an embedded 2-handle with suitable properties cannotbefoundinEliashberg’sconstruction. Toobtainthecorrectmanifold onemustperformaninductiveprocedurewhichcancels allsuperfluousloops caused by kinks, thereby creating Casson handles which are homeomorphic, but not diffeomorphic, to the standard index two handle D2 ×D2 (Freed- man [25]). In §7 we follow a similar path to construct a holomorphic map in the chosen homotopy class, performing the Casson tower construction simultaneously at a possibly increasing number of places. Organization of the paper. In §2 we recall the relevant notions from Stein and contact geometry. Sections §3 – §5 contain preparatory lemmas. The main geometric ingredient is Lemma 3.1 which gives totally real discs at- tachedfromtheexteriortoastronglypseudoconvexdomainalongacomplex tangential sphere. A main analytic ingredient is an approximation theorem for holomorphic maps to arbitrary complex manifolds (Theorem 4.1). Lemma 5.1 provides an approximate extension of a holomorphic map to an attached handle. The main results are presented and proved in sections §6 (for dimCX 6= 2) and §7 (for dimCX = 2). 2. Preliminaries We begin by recalling some basic notions of the handlebody theory; see e.g. [26], [29], [46]. Let X be a smooth compact n-manifold with boundary ∂X, and let Dk denote the closed unit ball in Rk. A k-handle H attached to X is acopy of Dk×Dn−k smoothly attached to∂X along ∂Dk×Dn−k, with thecornerssmoothed,whichgivesalargercompactmanifoldwithboundary. The central disc Dk × {0}n−k is the core of H. A handle decomposition of a smooth (open or closed) manifold X is a representation of X as an increasing union of compact domains with boundary X ⊂ X such that j X isobtainedbyattaching ahandletoX . (Inthecaseofopenmanifolds j+1 j one takes the interior of the resulting handlebody.) By Morse theory every smooth manifold admits a handlebody representation. An almost complex structure on an even dimensional smooth manifold X is a smooth endomorphism J ∈ EndR(TM) satisfying J2 = −Id. The 6 FRANCFORSTNERICˇ &MARKOSLAPAR operator J gives rise to the conjugate differential dc, defined on functions by hdcρ,vi = −hdρ,Jvi for v ∈ TX, and the Levi form operator ddc. J is said tobeintegrableifeverypointofX admitsanopenneighborhoodU ⊂ X and a J-holomorphic coordinate map of maximal rank z = (z ,...,z ): U → Cn 1 n (n = 1dimRX), i.e., satisfying dz ◦J = idz; for a necessary and sufficient 2 integrability condition see Newlander and Nirenberg [48]. If h: X → X′ is a diffeomorphism and J′ is an almost complex structure on X′, we denote by J = h∗(J′) the (unique) almost complex structure on X satisfyingdh◦J = J′◦dh; i.e., suchthathis abiholomorphism. Similarly we denote by J′ = h (J) the push-forward of an almost complex structure ∗ J by h. A map f′: X′ → Y to a complex manifold Y is J′-holomorphic if and only if f = f′◦h: X → Y is J-holomorphic with J = h∗(J′). An integrable structure J on a smooth manifold X is said to be Stein if (X,J) is a Stein manifold; this is the case if and only if there is a strongly J- plurisubharmonic Morse exhaustion functionρ: X → R,i.e.,hddcρ,v∧Jvi > 0 for every 0 6= v ∈ TX (Grauert [33]). The (1,1)-form ω = ddcρ = 2i∂∂ρ is then a symplectic form on X, defining a J-invariant Riemannian metric g(v,w) = hω,v ∧Jwi (v,w ∈ TX). The Morse indices of such function ρ are ≤ n = 1 dimRX and hence X is the interior of a handlebody without 2 handles of index > n [1, 46]. A real subbundle V of the tangent bundle TX is said to be J-real, or C totally real, if V ∩JV = {0} for every x ∈ X; its complexification V = x x V ⊗RC can be identified with the J-complex subbundleV ⊕JV of TX. An immersion G: D → X of a smooth manifold D into X is J-real (or totally real) if dG (T D) is a J-real subspace of T X for every x∈ D. x x G(x) Let W be a relatively compact domain with smooth boundary Σ = ∂W in an almost complex manifold (X,J). The set ξ = TΣ∩J(TΣ) is a corank one J-complex linear subbundle of TΣ. Assume now that ρ is a smooth function in a neighborhood of Σ=∂W such that Σ ={ρ = 0}, dρ 6= 0 on Σ and ρ < 0 on W. Let η := dcρ| , a one-form on Σ with kerη = ξ. We say TΣ that Σ is strongly J-pseudoconvex, or simply J-convex, if hddcρ,v∧Jvi > 0 for all 0 6= v ∈ ξ; this condition is independent of the choice of ρ. (We shall omitJ whenitis clear whichalmostcomplexstructuredowehaveinmind.) This implies that η∧(dη)n−1 6= 0 on Σ (n = dimCX) which means that η is a contact form and (Σ,ξ) is a contact manifold (see [6, 7, 36, pp. 338–340]). A smooth function ρ: X → R whose level sets are J-convex outside of the critical points is said to be J-convex. An immersion g: S → Σ of a smooth manifold S into a contact manifold (Σ,ξ) is Legendrian if dg(TS) ⊂ ξ. In the case at hand, when Σ is the boundary of a strongly pseudoconvex domain, another common expression is a complex tangential immersion. Let J denote the standard complex structure on Cn. For a fixed k ∈ st {1,...,n} let z = (z ,...,z ) = (x′ + iy′,x′′ + iy′′), with z = x + iy , 1 n j j j STEIN STRUCTURES AND HOLOMORPHIC MAPPINGS 7 denote the coordinates on Cn corresponding to the decomposition Cn = Ck ⊕Cn−k = Rk ⊕iRk ⊕Rn−k ⊕iRn−k. Let D = Dk ⊂ Rk be the closed unit ball in Rk and S = Sk−1 = ∂D its boundary sphere. Identifying Dk with its image in Rk ⊕{0}2n−k ⊂ Cn we obtain the core of the standard index k handle (2.1) H = (1+δ)Dk ×δD2n−k ⊂ Cn, δ > 0. δ A standard handlebody of index k in Cn is a set K = Q ∪H for some λ,δ λ δ 0< λ < 1 and 0 < δ < 2λ (Fig. 1), where 1−λ (2.2) Q = z = (x′+iy′,z′′)∈ Ck ⊕Cn−k: |y′|2+|z′′|2 ≤ λ(|x′|2−1) . λ (cid:8) (cid:9) The condition λ < 1 insures that Q is strongly pseudoconvex, and the λ bound on δ implies (1+δ)∂Dk ×δD2n−k ⊂ Q . λ We shall need the following result of Eliashberg [7, §3]. (See also [22].) Lemma 2.1. (Eliashberg) For every ǫ > 0 and λ > 1 there exist a number δ ∈ (0,ǫ) and a smoothly bounded, strongly pseudoconvex handlebody L ⊂ Cn with core Q ∪Dk such that K ⊂ L ⊂ K (Fig. 1). λ λ,δ λ,ǫ core Dk H δ H ǫ Qλ b Qλ L Figure 1. A strongly pseudoconvex handlebody L Eliashberg’s construction gives an L of the form L = (x′+iy′,z′′)∈ Cn: |y′|2+|z′′|2 ≤ h(|x′|2) (cid:8) (cid:9) whereh: [0,∞] → [δ2,∞] is asmooth, increasing, convex function chosen so that L is a tubeof constant radius δ around Dk ⊂ Cn over a slightly smaller ball rDk (r < 1), and L equals Q over r′Dk for some r′ > 1 close to 1. λ 8 FRANCFORSTNERICˇ &MARKOSLAPAR We introduce the following (trivial) bundles over the disc D ⊂ Rk ⊕ {0}2n−k: ∂ ∂ ν′ = Span ,..., = D× {0}k ⊕Rk ⊕{0}2n−2k , n o(cid:12) ∂y1 ∂yk (cid:12)D (cid:0) (cid:1) (cid:12) ∂ ∂ ν′′ = Span , : j = k+1,...,n = D× {0}2k ⊕R2n−2k , n o(cid:12) ∂xj ∂yj (cid:12)D (cid:0) (cid:1) (cid:12) ν = ν′⊕ν′′ = D× {0}k ⊕R2n−k . (cid:0) (cid:1) Thus ν′ = J (TD), TCD = TD⊕ν′, and TCn| = TD⊕ν = TCD⊕ν′′. st D Let v → S denote the (trivial) real line bundle over S spanned by the vector field k x ∂ . Over S we then have further decompositions Pj=1 j∂xj TD| = v⊕TS, ν′| = J (v)⊕J (TS), TD| ⊕ν′| ≃ vC⊕TCS. S S st st S S C Note that T S is a trivial complex vector bundle. Given a smooth embedding (or immersion) G: D → X of the disc D = Dk ⊂ Cn to a smooth 2n-dimensional manifold X, a normal framing over G is a homomorphism β: ν → TX| such that dG ⊕β : T D ⊕ν = G(D) x x x x T Cn → T X is a linear isomorphism for every x∈ D. x G(x) 3. Totally real discs attached to strongly pseudoconvex domains along Legendrian spheres Let W be an open, relatively compact domain with smooth strongly pseudoconvex boundary Σ = ∂W in an almost complex manifold (X,J) of real dimension 2n. Choose a smooth defining function ρ for W which is strongly J-plurisubharmonic near Σ = {ρ = 0}. Let w ⊂ TX| be the Σ orthogonal complement of TΣ with respect to the metric associated to the symplectic form ddcρ (see §2); thus w is spanned by the gradient of ρ with respect to this metric. Then Jw ⊂ TΣ and we have orthogonal decompositions TX| = w⊕TΣ= w⊕Jw⊕ξ, where ξ = TΣ∩J(TΣ). Σ Let D = Dk, S = Sk−1 = ∂D and v be as in §2. An embedding of a pair G: (D,S) → (X\W,Σ) is a smooth embedding G: D ֒→ X\W such that G(S) = G(D) ∩ Σ and G is transverse to Σ along G(S). Such G is said to be normal to Σ if dG (v ) = w for every x ∈ S, i.e., G maps x x G(x) the direction orthogonal to S ⊂ Rk into the direction orthogonal to Σ ⊂ X. The analogous definition applies to immersions. Thefollowing lemmaisakey geometric ingredientintheproofof allmain results in this paper. Its proof closely follows the construction of a special handle attaching triple (HAT) in §2 of Eliashberg’s paper [7], but with an additional argument in the critical case k = n 6= 2 (see Remark 3.2 below). We thank Y. Eliashberg for his help in the proof of the critical case (private communication, June 2005). STEIN STRUCTURES AND HOLOMORPHIC MAPPINGS 9 Lemma 3.1. Let W be an open, relatively compact domain with smooth strongly pseudoconvex boundary Σ = ∂W in an almost complex manifold (X,J). Let 1 ≤ k ≤ n = 1 dimRX, D = Dk, S = ∂D. Given a smooth em- 2 bedding G : (D,S) → (X\W,Σ), there is a regular homotopy of immersions 0 G : (D,S) → (X\W,Σ) (t ∈ [0,1]) which is C0 close to G such that the im- t 0 mersion G : D → X\W is J-real and normal to Σ, and g := G | : S ֒→ Σ 1 1 1 S is a Legendrian embedding. If k < n, or if k = n 6= 2, there exists an isotopy of embeddings G with these properties. If J is integrable in a neighborhood t of Σ∪G (D) and Σ is real-analytic then G can be chosen real analytic. 0 1 As was pointed out in [7, Note 2.4.2.], the topological obstruction in the case k = n = 2 is essential. For example, there does not exist an embedded totally real 2-disc in C2\B, attached to the ball B ⊂ C2 along a Legendriancurvein∂B,sinceby[7]theresultingconfigurationwouldadmit an open Stein neighborhood diffeomorphic to S2 × R2 in contradiction to the generalized adjunction inequality (1.1). Proof. The scheme of proof is illustrated on Fig. 2. First we find a regular homotopy from the initial disc G : D ֒→ X\W to an immersed disc 0 G : D → X\W which is attached with a correct normal framing to ∂W 1 along an embedded Legendrian sphere. Next we deform G by a regular 1 homotopy which is fixed near the boundary to a totally real immersed disc G , using the h-principle for totally real immersions. Finally we show that, 2 unless k = n = 2, the construction can be done by isotopies of embeddings. G 1 g b 1 g b 0 W G G 0 2 g 0 b g 1 b Figure 2. Deformations of an attached disc Set g = G | : S ֒→ ∂W. By a correction of G along S (keeping g 0 0 S 0 0 fixed) we may assume that it is normal to Σ, i.e., such that l := dG | 0 0 v maps v to w| . Choose a complex vector bundle isomorphism g0(S) φ : TCn| = D×Cn → TX| , φ ◦J = J ◦φ 0 D G0(D) 0 st 0 10 FRANCFORSTNERICˇ &MARKOSLAPAR covering G . We shall use the coordinates on Cn introduced in §2. The 0 vector field τ = k x ∂ is outer radial to the sphere S = ∂D in Pj=1 j∂xj Rk × {0}2n−k. Let τ be the unique nonvanishing vector field on Cn over S satisfying φ (τ ) = ℓ (τ ) for every x ∈ S. By dimension reasons there 0 x e 0 x exists a map A: D → GL (C) satisfying A τ = τ for x∈ S. Replacing φ e n x x x 0 by φ ◦A we may (and shall) assume from now on that φ | = ℓ . A further 0 e 0 v 0 homotopic correction of φ insures that φ (TCS ⊕ν′′| ) = ξ| , thereby 0 0 S g0(S) providing a trivialization of the latter bundle. Write φ0 = φ′0 ⊕ φ′0′ where φ′0 = φ0|TCD and φ′0′ = φ0|ν′′ (we use the notation of §2). Setting ψ0 := φ0|TCS we thus have φ′0|TCD|S = ℓC0 ⊕ψ0: vC ⊕TCS → TX|g0(S) = wC ⊕ξ|g0(S). Note that ψ ⊕φ′′: TCS⊕ν′′| → ξ| is a complex vector bundleisomor- 0 0 S g0(S) phism. Furthermore, there is a homotopy of real vector bundle monomor- phisms ι : TD ֒→ TX| (s ∈ [0,1]) satisfying s G0(D) ι = dG , ι = φ | , ι | = ℓ : v → w| (s ∈ [0,1]). 0 0 1 0 TD s v 0 g0(S) Considerthepair(g ,ψ )consistingof theembeddingg : S ֒→ Σandthe 0 0 0 C-linearembeddingψ : TCS ֒→ ξ| ofthecomplexifiedtangentbundleof 0 g0(S) S (a trivial complex vector bundleof rank k−1) into the contact subbundle ξ ⊂ TΣ over the map g . By the Legendrization theorem of Gromov ([36], p. 0 339, (B’)) andDuchamp [6]thereexists aLegendrianembeddingg : S ֒→ Σ 1 C whosecomplexifieddifferentialψ := d g ishomotopictoψ byahomotopy 1 1 0 of C-linear vector bundle embeddings ψ : TCS ֒→ ξ (t ∈ [0,1]). t Let Hom (TS,TΣ) denote the space of all fiberwiseinjective real vector inj bundlemapsTS ֒→ TΣ. ConsiderthepathinHom (TS,TΣ)beginningat inj dg and ending at dg , consisting of the homotopy ι | (s ∈ [0,1]) followed 0 1 s TS by the homotopy ψ | (t ∈ [0,1]) (left and top side of the square in Fig. t TS 3). By Hirsch’s one parametric h-principle for immersions [36, 39] this path can be deformed in the space Hom (TS,TΣ) (with fixed ends) to a path inj dg : TS ֒→ TΣ| where g : S → Σ (t ∈ [0,1]) is a regular homotopy of t gt(S) t immersions from g to g . We can insure that ψ covers the base map g for 0 1 t t all t ∈ [0,1]. This gives a two parameter homotopy θ ∈ Hom (TS,TΣ) t,s inj for (t,s) ∈[0,1]2 satisfying the following conditions (Fig. 3): (i) θ = dg (bottom side), t,0 t (ii) θ = ψ | (top side), t,1 t TS (iii) θ = ι | (left side; hence θ = dg and θ = ψ | ), 0,s s TS 0,0 0 0,1 0 TS (iv) θ = dg (right side), and 1,s 1 (v) θ covers g for every t,s ∈ [0,1]. t,s t We can extend g to a regular homotopy of immersions G : (D,S) → t t (X\W,Σ) (t ∈ [0,1]) which are normal to Σ, beginning at t = 0 with the given map G . Let ℓ := dG | : v → w| . By the homotopy lifting 0 t t v gt(S)