Stein structures and holomorphic mappings

0.45 MB·English

by  Franc Forstneric

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 com-

pact 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 cen-
tral 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 com-
plex 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 ‘holomor-
phic 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 intro-
duced (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 sur-
face, 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 theo-
rem 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 decomposi-
tions 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 reg-
ular 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)