arXiv:0704.0124v3  [math.CV]  5 Oct 2008
Proper J-holomorphic discs in Stein domains of
dimension 2
Bernard Coupet*, Alexandre Sukhov** and Alexander Tumanov***
* Université de Provence, CMI, 39 rue Joliot-Curie, Marseille, Cedex, Bernard.Coupet@cmi.
math-mrs.fr
** Université des Sciences et Technologies de Lille, Laboratoire Paul Painlevé, U.F.R. de Mathé-
matique, 59655 Villeneuve d’Ascq, Cedex, France, sukhov@math.univ-lille1.fr
*** University of Illinois, Department of Mathematics 1409 West Green Street, Urbana, IL
61801, USA, tumanov@illinois.edu
Abstract. We prove the existence of global Bishop discs in a strictly pseudoconvex Stein
domain in an almost complex manifold of complex dimension 2.
MSC: 32H02, 53C15.
Key words: almost complex manifold, strictly pseudoconvex domain, Morse function,
Bishop disc.
1 Introduction
The problem of embedding complex discs or general Riemann surfaces into complex manifolds
has been well-known for a long time. The interest to the case of almost complex manifolds
has grown due to a strong link with symplectic geometry (Gromov [13]). We present the
following result.
Theorem 1.1 Let (M,J) be an almost complex manifold of complex dimension 2 admitting
a strictly plurisubharmonic exhaustion function ρ. Then for every non-critical value c of ρ,
every point p ∈ Ωc = {ρ < c} and every vector v ∈ Tp(M) there exists a J-holomorphic
immersion f : ID −→ Ωc, where ID ⊂ IC is the unit disc, such that f(bID) ⊂ bΩc, f(0) = p,
and df0
∂Re ζ
= λv for some λ > 0.
For a domainM ⊂ ICn with the standard complex structure, the result is due to Forstnerič
and Globevnik [12]; there are various generalizations including embedding bordered Riemann
surfaces into singular complex spaces (see [7] and references there).
http://arxiv.org/abs/0704.0124v3
Recently Biolley [4] proved a similar result for an almost complex manifold M of any
dimension n, but under the additional hypothesis that the defining function ρ is subcritical.
The latter means that ρ does not have critical points of the maximum Morse index n. (A
plurisubharmonic function can not have critical points of index higher than n.) We don’t
impose such a restriction. Furthemore, Biolley [4] does not prescribe the direction of the disc.
Her method is based on the Floer homology and substantially uses recent work of Viterbo
[23] and Hermann [14]. Our proof is self-contained; we adapt the ideas of Forstnerič and
Globevnik [12] to the almost complex case using the methods of classical complex analysis
and PDE.
In most work on the existence of global discs with boundaries in prescribed totally real
manifolds ([2, 9, 10, 15, 17] and others) the authors use the continuity principle. By the
implicit function theorem and the linearized equation they show that any given disc generates
a family of nearby discs. Then the compactness argument allows for passing to the limit. In
contrast, we construct the discs by solving the almost Cauchy-Riemann equation directly.
Following [12], we start with a small disc passing through the given point in given direction
and push the boundary of the disc in the directions complex-tangent to the level sets of the
defining function ρ; it results in increasing ρ due to pseudoconvexity. This plan leads to
a problem of attaching J-holomorphic discs to totally real tori in a level set of ρ. The
problem is of independent interest and may occur elsewhere. It reduces in turn to the
existence theorem for a boundary value problem for a quasilinear elliptic system of partial
differential equations in the unit disc (Theorem 4.1). We prove it by the classical methods of
the Beltrami equations and quasiconformal mappings (Ahlfors, Bers, Boyarskii, Lavrentiev,
Morrey, Vekua; see [3, 21] and references there). The result can be viewed as a far reaching
generalization of the Riemann mapping theorem.
Since the almost Cauchy-Riemann equation is nonlinear, one can only hope to find a
solution close to a current disc f . By measuring the closeness in the Lp norm, we are able in
fact to construct a disc sufficiently far from f in the sup-norm. To make sure we are looking
for a disc close to f , we adapt the idea of [12] of adding to f(ζ) a term with a factor of ζn
(ζ ∈ ID) with big n. We develop a nonlinear version of this idea.
The above procedure works well in the absence of critical points of ρ. In order to push
the boundary of the disc through critical level sets, we use a method by Drinovec Drnovšek
and Forstnerič [7, 11], which consists of temporarily switching to another plurisubharmonic
function at each critical level set. We point out that adapting this method to the almost
complex case is not a major problem because the difficulties are localized near the critical
points, in which the almost complex structure can be closely approximated by the standard
complex structure.
Although higher dimension gives one more freedom for constructing J-holomorphic discs,
we must admit that our proof of the main result goes through in dimension 2 only. The reason
is that our main tool (Theorem 4.1) needs a special coordinate system in which coordinate
hyperplanes z = const are J-complex, which generally can be achieved only in dimension
2. For a domain in ICn with the standard complex structure, the result is obtained in [12]
by reduction to dimension 2 using sections by 2-dimensional complex hypersurfaces. Such a
reduction in not possible for almost complex structures.
We thank Franc Forstnerič and Josip Globevnik for helpful discussions, in particular, for
pointing out at some difficulties in the problem and for the important references [7, 11].
Parts of the work were completed when the third author was visiting Université de
Provence and Université des Sciences et Technologies de Lille in the spring of 2006. He
thanks these universities for support and hospitality.
2 Almost complex manifolds
Let (M,J) be an almost complex manifold. Denote by ID the unit disc in IC and by Jst the
standard complex structure of ICn; the value of n is usually clear from the context. Let f be a
smooth map from ID intoM . Recall that f is called J-holomorphic if df ◦Jst = J◦df . We also
call such a map f a J-holomorphic disc or a pseudoholomorphic disc or just a holomorphic
disc when a complex structure is fixed. We will often denote by ζ the standard complex
coordinate on IC.
A fundamental result of the analysis and geometry of almost complex structures is the
Nijehnuis–Woolf theorem which states that given point p ∈ M and given tangent vector
v ∈ TpM there exists a J-holomorphic disc f : ID −→M centered at p, that is, f(0) = p and
such that df(0)(∂/∂Re ζ) = λv for some λ > 0. This disc f depends smoothly on the initial
data (p, v) and the structure J . A short proof of this theorem is given in [19]. This result
will be used several times in the present paper.
It is well known that an almost complex manifold (M,J) of complex dimension n can be
locally viewed as the unit ball IB in ICn equipped with an almost complex structure which
is a small deformation of Jst. More precisely, let (M,J) be an almost complex manifold of
complex dimension n. Then for every p ∈M , δ0 > 0, and k ≥ 0 there exist a neighborhood U
of p and a smooth coordinate chart z : U −→ IB such that z(p) = 0, dz(p) ◦ J(p) ◦ dz−1(0) =
Jst, and the direct image z∗(J) := dz ◦ J ◦ dz
−1 satisfies the inequality ||z∗(J)− Jst||Ck(ĪB) ≤
δ0. For a proof we point out that there exists a diffeomorphism z from a neighborhood
U ′ of p ∈ M onto IB such that z(p) = 0 and dz(p) ◦ J(p) ◦ dz−1(0) = Jst. For δ > 0
consider the isotropic dilation dδ : t 7→ δ
−1t in ICn and the composite zδ = dδ ◦ z. Then
limδ→0 ||(zδ)∗(J)−Jst||Ck(ĪB) = 0. Setting U = z
δ (IB) for positive δ small enough, we obtain
the desired result. As a consequence we obtain that for every point p ∈ M there exists a
neighborhood U of p and a diffeomorphism z : U → IB with center at p (in the sense that
z(p) = 0) such that the function |z|2 is J-plurisubharmonic on U and z∗(J) = Jst +O(|z|).
Let u be a function of class C2 on M , let p ∈ M and v ∈ TpM . The Levi form of u at p
evaluated on v is defined by LJ(u)(p)(v) := −d(J∗du)(v, Jv)(p).
The following result is well known (see, for instance, [6]).
Proposition 2.1 Let u be a real function of class C2 on M , let p ∈M and v ∈ TpM . Then
LJ(u)(p)(v) = ∆(u◦f)(0) where f : rID −→M for some r > 0 is an arbitrary J-holomorphic
map such that f(0) = p and df(0)(∂/∂Re ζ) = v, ζ ∈ rID.
The Levi form is invariant with respect to J-biholomorphisms. More precisely, let u be a
C2 real function onM , let p ∈M and v ∈ TpM . If Φ is a (J, J
′)-holomorphic diffeomorphism
from (M,J) into (M ′, J ′), then LJ(u)(p)(v) = LJ
(u ◦ Φ−1)(Φ(p))(dΦ(p)(v)).
Finally, it follows from Proposition 2.1 that a C2 function u is J-plurisubharmonic on
M if and only if LJ (u)(p)(v) ≥ 0 for all p ∈ M , v ∈ TpM . Thus, similarly to the case of
the integrable structure one arrives in a natural way to the following definition: a C2 real
valued function u on M is strictly J-plurisubharmonic on M if LJ(u)(p)(v) is positive for
every p ∈M , v ∈ TpM\{0}.
Let J be a smooth almost complex structure on a neighborhood of the origin in ICn and
J(0) = Jst. Denote by z = (z1, ..., zn) the standard coordinates in IC
n (in matrix computations
below we view z as a column). Then a map z : ID −→ ICn is J-holomorphic if and only if it
satisfies the following system of partial differential equations
zζ − A(z)zζ = 0, (1)
where A(z) is the complex n× n matrix defined by
A(z)v = (Jst + J(z))
−1(Jst − J(z))v (2)
It is easy to see that right-hand side of (2) is IC-linear in v ∈ ICn with respect to the standard
structure Jst, hence A(z) is well defined. Since J(0) = Jst, we have A(0) = 0. Then in a
sufficiently small neighborhood U of the origin the norm ‖ A ‖L∞(U) is also small, which
implies the ellipticity of the system (1).
However, we will need a more precise choice of coordinates imposing additional restric-
tions on the matrix function A. The proof of the following elementary statement can be
found, for instance, in [6].
Lemma 2.2 After a suitable polynomial second degree change of local coordinates near the
origin
z 7→ z +
akjzkzj
we can achieve
A(0) = 0, Az(0) = 0
In these coordinates the Levi form of a given C2 function u with respect to J at the origin
coincides with its Levi form with respect to Jst that is
LJ(u)(0)(v) = LJst(u)(0)(v)
for every vector v ∈ T0IR
3 Integral transforms in the unit disc
Let Ω be a domain in IC. Let TΩ denote the Cauchy-Green transform
TΩf(ζ) =
f(τ)dτ ∧ dτ
τ − ζ
. (3)
Let RΩ denote the Ahlfors-Beurling transform
RΩf(ζ) =
f(τ)dτ ∧ dτ
(τ − ζ)2
, (4)
where the integral is considered in the sense of the Cauchy principal value. We omit the
index Ω if it is clear form the context. Denote by B the Bergman projection for ID.
Bf(ζ) =
f(τ)dτ ∧ dτ
(τζ − 1)2
We need the following properties of the above operators.
Proposition 3.1 (i) Let p > 2 and α = (p−2)/p. Then the linear operator T : Lp(ID) −→
Cα(IC) is bounded, in particular, T : Lp(ID) −→ L∞(ID) is compact. If f ∈ Lp(ID),
then ∂ζTf = f , ζ ∈ ID, as a Sobolev derivative.
(ii) Let m ≥ 0 be integer and let 0 < α < 1. Then the linear operators T : Cm,α(ID) −→
Cm+1,α(IC) and R : Cm,α(ID) −→ Cm,α(ID) are bounded. Furthermore,if f ∈ Cm,α(ID),
then ∂ζTf = f and ∂ζTf = Rf , ζ ∈ ID, in the usual sense.
(iii) The operator RΩ can be uniquely extended to a bounded linear operator RΩ : L
p(Ω) −→
Lp(Ω) for every p > 1. If f ∈ Lp(ID), p > 1 then ∂ζTf = Rf as a Sobolev derivative.
Moreover, the operator RIC is an isometry of L
2(IC), therefore ‖ RIC ‖L2(IC)= 1.
(iv) The Bergman projection B : Lp(ID) −→ Ap(ID) is bounded. Here Ap(ID) denotes the
space of all holomorphic functions in ID of class Lp(ID).
(v) The functions p 7→‖ T ‖Lp(Ω) and p 7→‖ R ‖Lp(Ω) are logarithmically convex and in
particular, continuous for p > 1.
The proofs of the parts (i)–(iii) are contained in [21]. The part (iv) follows from (iii); see e.
g. [8]. The part (v) follows by the classical interpolation theorem of M. Riesz–Torin (see e.
g. [24]).
We introduce modifications of the operators T and R for solving certain boundary value
problems in the unit disc ID. For f ∈ Lp(ID) we define
T0f(ζ) = Tf(ζ)− Tf(ζ
), ζ ∈ ID. (5)
By Proposition 3.1 for p > 2 and α = (p− 2)/p, the linear operator T0 : L
p(ID) −→ Cα(ID)
is bounded, in particular, T0 : L
p(ID) −→ L∞(ID) is compact. Since the function Tf is
holomorphic and bounded in IC\ID, then the function ζ 7→ (Tf)(ζ
) is holomorphic in ID.
Hence ∂ζT0f = ∂ζTf = f . Furthermore, for ζ ∈ bID, we have ζ = ζ
, therefore by (5),
ReT0f(ζ) = 0. Hence for f ∈ L
p(ID), the function u = T0f solves the boundary value
problem
∂ζu = f, ζ ∈ ID,
Reu|bID = 0
We further define
R0f := ∂ζT0f.
Since ∂ζTf = Rf and ∂ζTf = f , then
R0f(ζ) = ∂ζT0f(ζ) = Rf(ζ)− ∂ζTf(ζ
) = Rf(ζ) + ζ−2Rf(ζ
), (6)
and we obtain a nice formula
R0f = Rf +Bf,
where B is the Bergman projection. By Propositions 3.1(iv) and (v), the operator R0 :
Lp(ID) −→ Lp(ID) is bounded, and the map p 7→‖ R0 ‖Lp(ID) is continuous for p > 1. By
Proposition 3.1(iii), R is an isometry of L2(IC). The analogue of this result for the operator
R0 may have been used for the first time by Vinogradov [22]. In fact we came across [22]
after proving the following
Theorem 3.2 R0 is a IR-linear isometry of L
2(ID), in particular, ‖ R0 ‖L2(ID)= 1.
Since we could not find a proof in the literature, for completeness we include it here.
Proof : For a domain G ⊂ IC we use the inner product
(f, g)G = −
fgdζ ∧ dζ.
We put
σf(ζ) = ζ
), ψ(ζ) = ζ
Then σ2 = id. By substitution ζ 7→ ζ
we obtain
(σf, σg)ID = (g, f)IC\ID, Rσ = ψσR, R = ψσRσ. (7)
By (6) we have
R0f = Rf + ψσRf.
Let f ∈ L2(ID). Extend f to all of IC by putting f(ζ) = 0 for |ζ | > 1. Then
‖ R0f ‖
L2(ID)= (Rf + ψσRf,Rf + ψσRf)ID =
(Rf,Rf)ID + 2Re (Rf, ψσRf)ID + (ψσRf, ψσRf)ID.
Since |ψ| = 1, by (7) we obtain
(ψσRf, ψσRf)ID = (σRf, σRf)ID = (Rf,Rf)IC\ID,
(Rf, ψσRf)ID = (ψσRσf, ψσRf)ID = (Rσf,Rf)IC\ID = (ψσRf,Rf)IC\ID = (Rf, ψσRf)IC\ID.
Then by the previous line and because R is an isometry
2Re (Rf, ψσRf)ID = Re (Rf, ψσRf)IC = Re (Rf,Rσf)IC = Re (f, σf)IC = 0.
Hence
‖ R0f ‖
L2(ID)= (Rf,Rf)ID + (Rf,Rf)IC\ID =‖ Rf ‖
L2(IC)=‖ f ‖
L2(IC)=‖ f ‖
L2(ID),
which proves the theorem.
4 Riemann mapping theorem for an elliptic system
The Riemann mapping theorem asserts that for every simply connected domain G ⊂ IC there
exists a conformal map of G onto ID. If G is smooth, then there is a diffeomorphism f :
G −→ ID, which defines an almost complex structure J = f∗(Jst) in ID. Then the Riemann
mapping theorem reduces to constructing a J-holomorphic map z : (ID, Jst) −→ (ID, J). The
latter satisfies the Beltrami type equation ∂ζz = A(z)∂ζz, which is equivalent to the linear
Beltrami equaion ∂zζ + A(z)∂zζ = 0. We consider the following more general system
∂ζz = a(z, w)∂ζz,
∂ζw = b(z, w)∂ζz,
which cannot be reduced to a linear one. Here z, w are unknown functions of ζ ∈ ID and a, b
are C∞ coefficients. By eliminating ζ , the system reduces to a nonhomogeneous quasilinear
Beltrami type equation ∂zw + a∂zw = b, but we prefer to deal with (8) directly.
The following theorem is our main technical tool for constructing pseudoholomorphic
discs with boundaries in a prescribed torus. For r > 0 denote IDr := rID.
Theorem 4.1 Let a, b : ID× ID1+γ −→ IC (γ > 0) be smooth functions such that
a(z, 0) = b(z, 0) = 0 and |a(z, w)| ≤ a0 < 1.
Then there exists C > 0 such that for every integer n ≥ 1 the system (8) admits a smooth
solution (zn, wn) with the following properties:
(i) |zn(ζ)| = |wn(ζ)| = 1 for |ζ | = 1.
(ii) zn : ID −→ ID is a diffeomorphism with zn(0) = 0.
(iii) |wn(ζ)| ≤ C|ζ |
n, |wn(ζ)| < 1 + γ.
Proof : Shrinking γ > 0 if necessary, we extend the functions a and b to all of IC2
preserving their properties. We will look for a solution of (8) in the form
z = ζeu, w = ζnev.
Then for the new unknowns u and v we have the following boundary value problem
∂ζu = A(u, v, ζ)(1 + ζ∂ζu), ζ ∈ ID
∂ζv = B(u, v, ζ)(1 + ζ∂ζu), ζ ∈ ID
Re u(ζ) = Re v(ζ) = 0, |ζ | = 1
where
A = aζ−1eu−u,
B = bζ−neu−v.
Put ∂ζu = h and choose u in the form u = T0h. Then ∂ζu = R0h, which we plug into (9).
We obtain the following system of singular integral equations for u, v and h:
h = A(1 + ζR0h),
u = T0h,
v = T0(B(1 + ζR0h))
We denote by ‖ f ‖p the L
p-norm of f in ID. Since the function p 7→‖ R0 ‖p is continuous in
p and ‖ R0 ‖2= 1 we choose p > 2 such that
a0 ‖ R0 ‖p< 1.
For given u, v ∈ L∞(ID) the map h 7→ A(1 + ζR0h) is a contraction in L
p(ID) because
‖ ζA ‖∞‖ R0 ‖p< 1.
Hence there exists a unique solution h = h(u, v) of the first equation of (10) satisfying
‖ h ‖p≤
‖ A ‖p
1− a0 ‖ R0 ‖p
Consider the map F : L∞(ID)× L∞(ID) −→ L∞(ID)× L∞(ID) defined by
F : (u, v) 7→ (U, V ) = (T0h, T0(B(1 + ζR0h)))
where h = h(u, v) is determined above. Then F is continuous (even Lipschitz) map. Let
E = {(u, v) ∈ L∞(ID)× L∞(ID) :‖ u ‖∞≤ u0, ‖ v ‖∞≤ v0}
We need the following
Lemma 4.2 There exist u0 > 0, v0 > 0 such that E is invariant under F .
Assuming the lemma, we prove the existence of the solution of (10). Indeed, since
T0 : L
p(ID) −→ L∞(ID) is compact for p > 2, then F : E −→ E is compact. Since
E is a bounded, closed and convex, then the existence of the solution of (10) follows by
Schauder’s principle.
Proof of Lemma 4.2 : Since a(z, 0) = b(z, 0) = 0, we have
|a(z, w)| ≤ C1|w|, |b(z, w)| ≤ C1|w|.
Here and below we denote by Cj constants independent of n. We have
|a| = |a(ζeu, ζnev)| ≤ C1e
‖v‖∞ |ζ |n,
‖ A ‖p=‖ aζ
−1 ‖p≤ C2 ‖ ζ
n−1 ‖p e
‖v‖∞ ≤ C3e
‖v‖∞n−1/p.
By (11), ‖ h ‖p≤ C4e
‖v‖∞n−1/p, hence
‖ U ‖∞≤ C5e
‖v‖∞n−1/p.
Similarly
|B| = |b(ζeu, ζnev)ζ−neu−v| ≤ C1e
‖u‖∞ ,
‖ B ‖∞≤ C1e
‖ V ‖∞≤ C7(‖ B ‖p + ‖ B ‖∞‖ h ‖p) ≤ C8e
‖u‖∞ .
Let δ = n−1/p. Then
‖ U ‖∞≤ C9δe
‖v‖∞ ,
‖ V ‖∞≤ C9e
Consider the system
u0 = C9δe
v0 , v0 = C9e
with the unknowns u0, v0. Then
u0 = C9δe
For small δ > 0 this equation has two positive roots. Let u0 = u0(δ) be the smaller root and
v0 = v0(δ) = C9e
u0. Now if ‖ u ‖∞≤ u0, ‖ v ‖∞≤ v0, then
‖ U ‖∞≤ C9δe
‖v‖∞ ≤ C9δe
v0 ≤ u0,
‖ V ‖∞≤ C9δe
‖u‖∞ ≤ C9δe
u0 ≤ v0
Hence E is invariant under F , which proves the lemma.
Thus the solution of (10) in L∞(ID) exists for n big enough. Since h ∈ Lp(ID), p > 2,
the second and the third equations of (10) imply that u, v ∈ Cα(ID), α = (p − 2)/p. Since
∂ζu = h ∈ L
p(ID) and ∂ζu = R0h ∈ L
p(ID) as Sobolev’s derivatives, then u and v are
solutions of (9), hence z = ζeu and w = ζnev are solutions of (8). By the ellipticity of
the system, z, w ∈ C∞(ID). The smoothness up to the boundary can be derived directly
from the properties of the Beltrami equation; it also follows by the reflection principle for
pseudoholomorphic discs attached to totally real manifolds (see, e.g., [18]).
Since the winding number of z|bID about 0 equals 1 and
∣∂ζz/∂ζz
∣ = |a| ≤ a0 < 1 then
z : ID −→ ID is a homeomorphism by the classical properties of the Beltrami equation [21],
and (ii) follows.
Note that u0 −→ 0, v0 −→ C9 as n −→ ∞. Since T0 : L
p(ID) −→ Cα(ID) is bounded,
then we have
‖ v ‖Cα(ID)≤ C10, ‖ e
v ‖Cα(ID)≤ C11,
and |w(ζ)| ≤ C11|ζ |
n. Furthermore, since |ev| = 1 on bID, then |ev(ζ)| ≤ 1 + C11(1 − |ζ |)
for |ζ | < 1. Then |w(ζ)| ≤ |ζ |n(1 + C11(1 − |ζ |)
α), hence ‖ w ‖∞−→ 1 as n −→ ∞. Hence
‖ w ‖∞< 1+ γ for n big enough, and (iii) follows. This completes the proof of Theorem 4.1.
5 Pseudoholomorphic discs attached to real tori
This section concerns the geometrization of Theorem 4.1. We apply Theorem 4.1 in order
to obtain a crucial technical result on (approximately) attaching pseudoholomorphic discs
to a given real 2-dimensional torus in (M,J). We will use this result later for pushing discs
across level sets of the defining function ρ in Theorem 1.1.
The tori and the discs considered in this section are not arbitrary. We study a special
case which will suffice for the proof of the main result. Given a psedoholomorphic immersed
disc f , we associate with f a real 2-dimensional torus Λ formed by the boundary circles of
discs hζ centered at the boundary points f(ζ), ζ ∈ bID. Thus, our initial data is a pair (f,Λ).
Our goal is to construct a pseudoholomorphic disc with the boundary attached to the torus
Λ. First we find a suitable neighborhood of the disc f which can be parametrized by the
bidisc in IC2. We transport the structure J onto this bidisc and choose the coordinates there
such that the equations for J-holomorphic discs take the form used in Theorem 4.1. The
theorem will provide a pseudoholomorphic disc approximately attached to Λ.
5.1 Admissible parametrizations by the bidisc and generated tori
Let f : ID −→ (M,J) be a J-holomorphic disc of class C∞(ID). Suppose f is an immersion.
Let γ > 0. Given ζ ∈ D consider a J-holomorphic disc
hζ : (1 + γ)ID −→ M
satisfying the condition hζ(0) = f(ζ) and such that the direction dhζ(0)(
∂Re τ
) is not tangent
to f . Admitting some abuse of notation, we sometimes write hf(ζ) for hζ .
This allows to define a C∞ map
H : ID× (1 + γ)ID −→M, H(ζ, τ) = hζ(τ).
Then H has the following properties:
(i) For every ζ ∈ ID the map hζ := H(ζ, •) is J-holomorphic.
(ii) For every ζ ∈ ID we have H(ζ, 0) = f(ζ).
(iii) For every ζ ∈ ID the disc hζ is transversal to f at the point f(ζ).
We assume in addition that
(iv) H : ID× (1 + γ)ID −→M is locally diffeomorphic.
Then Λ = H(bID×(1+γ)bID) is a real 2-dimensional torus immersed intoM . It is formed
by a family of topological circles γζ = hζ((1 + γ)bID) parametrized by ζ ∈ bID. Every such a
circle bounds a J-holomorphic disc hζ : (1 + γ)ID −→ M centered at f(ζ). In particular the
torus Λ can be continuously deformed to the circle f(bID).
If the above conditions (i) - (iv) hold we say that a mapH is an admissible parametrization
of a neighborhood of f(ID) and Λ is the torus generated by H .
5.2 Ellipticity of admissible parametrizations
We prove the following consequence of Theorem 4.1.
Theorem 5.1 Let f : ID −→ (M,J) be a C∞ immersion J-holomorphic in ID. Suppose
that there exists an admissible parametrization H of a neighborhood of f(ID) and let Λ be
the generated torus. Then there exists an immersed J-holomorphic disc f̃ of class C∞(ID)
centered at f(0), tangent to f at f(0) and satisfying the boundary condition f̃(bID) ⊂ H(bID×
bID).
We stress that the boundary of f̃ is attached to the torus H(bID×bID) and not to Λ. However
since γ > 0 can be chosen arbitrarily close to 0, this leads to the following result sufficient
for applications.
Corollary 5.2 In the hypothesis of the former theorem for any positive integer n there exists
an immersed J-holomorphic disc fn of class C∞(ID) centered at f(0), tangent to f at f(0)
and such that dist(fn(bID),Λ) −→ 0 as n −→ ∞.
Here dist denotes any distance compatible with the topology of M .
We begin the proof of Theorem 5.1 with the remark that the discs hζ , ζ ∈ D, fill a subset
V of M containing f(ID) which can be viewed as a fiber space with the base f(ID) and the
generic fiber hζ((1 + γ)ID). Therefore the defined above map
H : ID× (1 + γ)ID −→ V
gives a natural parametrization of V by the bidisc Uγ := ID × (1 + γ)ID. Since H is locally
diffeomorphic (see (iv) above) the inverse map H−1 is defined in a neighborhood of every
point of V . This allows to define the almost complex structure J̃ = H∗(J) = dH−1 ◦ J ◦ dH
on Uγ . The structure J̃ has a special form. Indeed, in the standard basis of IR
4 we have
J̃11 J̃12
J̃21 J̃22
where J̃kj are real 2×2 matrices. We recall that in this basis the standard complex structure
st of IC has the form
It follows by the property (i) of H that the maps τ 7→ (c, τ) are J̃-holomorphic for every
fixed c. This implies that J̃12 = 0 and J̃22 = J
st . Furthermore, since the map ζ 7→ (ζ, 0) is
J̃-holomorphic, we have J̃11(z, 0) = J
st and J̃21(z, 0) = 0.
Let now g : ID −→ Uγ be a J̃ -holomorphic map. If we set ζ = ξ+iη, the Cauchy–Riemann
equations have expressing the J̃-holomorphicity of g have the form
= 0 (13)
Suppose now that the matrix Jst+J is invertible. Then the Cauchy–Riemann equations can
be rewritten in the form
gζ + A(g)gζ = 0 (14)
where A is defined by (2). If we use the notation g = (z, w), then the Cauchy–Riemann
equations (14) can be written in the form
∂ζz = a(z, w)∂ζz,
∂ζw = b(z, w)∂ζz
identical to (8). Furthermore, since J̃(z, 0) = Jst, the conditions a(z, 0) = b(z, 0) = 0 are
satisfied.
Proposition 5.3 We have ‖ a ‖∞< 1.
Proof : The proof consists of two steps. First we study the matrix J̃+Jst which determines
the matrix A in the Cauchy–Riemann equations (14).
Lemma 5.4 The matrix J̃(z, w) + Jst is non-degenerate for any (z, w) ∈ ID× (1 + γ)ID.
Proof : It suffices to verify the condition det(J̃11(z, w) + J
st ) 6= 0. For every fixed (z, w)
the matrix J̃11(z, w) defines a complex structure on the euclidean space IR
2 so there exists a
matrix P = P (z, w) such that
J̃11(z, w) = PJ
−1. (16)
Recall that the manifold J2 of all complex structures on IR
2 can be identified with the
quotient GL(2, IR)/GL(1, IC) and has two connected components: J +2 and J
2 . A structure
J̃11 belongs to J
2 (resp. to J
2 ) if in the representation (16) we have detP > 0 (resp.
detP < 0). Suppose now that det(PJ
st ) = 0 or equivalently det(PJ
st +J
st P ) =
0 at some point (z, w). If we denote by pjk the entries of the matrix P , the last equality means
jk=1 p
jk = 0 which together with the non-degeneracy of P implies that detP < 0 so
that J̃11(z, w) ∈ J
2 . On the other hand, for the point (z, 0) we have detP > 0 since
J̃11(z, 0) = J
st so J̃(z, 0) ∈ J
2 . But we can join the points (z, 0) and (z, w) by a real
segment, so this contradiction proves lemma.
Now we can conclude the proof of Proposition 5.3. It follows by Lemma 5.4 that the
Cauchy–Riemann equations (13) can be written in the form (15) on ID × (1 + γ)ID. The
Cauchy–Riemann equations are elliptic at every point and this condition is independent of
the choice of the coordinates. The system (15) is ellipitic at a point (z, w) if and only if
|a(z, w)| 6= 1. Since a(z, 0) = 0 we obtain by connectedness that |a| < 1 on ID × (1 + γ)ID,
which concludes the proof.
Now Theorem 5.1 follows by Theorem 4.1.
5.3 Construction of an admissible parametrization with a pre-
scribed generated torus
So far we studied a situation where an admissible parametrization of a neighborhood of
an immersed J-holomorphic disc was given and proved the existence of discs with bound-
aries close to the generated torus. In the proof of our main result, we need an admissible
parametrization of a neighborhood of a J-holomorphic disc with a given generated torus.
Let f : ID −→ M be an immersed J-holomorphic disc of class C∞(ID). We extend f
smoothly to a neighborhood of ID. Let U be a small neighborhood of bID. For every point
f(ζ), ζ ∈ U , consider a J-holomorphic disc hζ : 2ID −→ M . Suppose that the map hζ
smoothly depends on ζ ∈ U . Thus we obtain a smooth map
H : bID × ID −→ M, H : (ζ, τ) 7→ hζ(τ).
Then Λ := H(bID × bID) is a real 2-dimensional torus. In order to construct an admissible
parametrization with the generated torus Λ we need to extend the map H from the cylinder
bID × ID to the bidisc ID× ID.
Definition 5.5 We call the described above torus Λ admissible. We further put Xζ :=
Xf(ζ) = dhζ(0)(
∂Re τ
) for every ζ ∈ U .
Theorem 5.6 Let f : ID −→ (M,J) be an immersed J-holomorphic disc of class C∞(ID).
Let Λ be an admissible torus. Then there is a sequence of admissible tori Λn converging to
Λ such that for every n there exists an immersed J-holomorphic disc fn of class C∞(ID)
centered at f(0), tangent to f at f(0) and satisfying the boundary condition fn(bID) ⊂ Λn.
Proof : Let Λ be an admissible torus and let X be the vector field given by Definition
5.5. In general it is impossible to extend X as a non-vanishing vector field transversal to
f(ID) at every point. However, for any integer (not necessarily positive) n we can consider
the discs hnζ : τ 7→ hζ(ζ
nτ), where ζ ∈ bID. Their tangent vectors at the points f(ζ) are equal
to Xnζ := ζ
nXζ, where by multiplying a vector by a complex number ζ
n we mean applying
the operator (Re ζ + (Im ζ)J)n. We need the following
Lemma 5.7 After a suitable choice of n the vector field Xnζ can be extended on the disc as
a nonvanishing field transversal to f at every point.
Proof : First we look for a global parametrization of a neighborhood of f(ID). Fix an
arbitrary vector field Y transversal to f(ID) at every point. By Nijenhuis - Woolf theorem
we obtain a family of J-holomorphic discs gz : w 7→ gz(w), z ∈ ID so that gz(0) = f(z) and
Xf(ζ) is tangent to gz. Then the map G : (z, w) 7→ gz(w) is a local diffeomorphism from
a neighborhood of ID × ID onto a neighborhood of f(ID) and G(z, 0) = f(z) so we can use
the coordinates (z, w). We pull back the vector field X by G−1 and consider the vector field
(G−1)∗(X) : ζ 7→ (G
−1)∗(Xζ). Let m be the winding number of the w-component of the
vector field (G−1)∗(X) when ζ runs along the circle bID. We set n = −m. Then the field
(G−1)∗(X
n) extends on the disc (ζ, 0) as a smooth vector field Z transversal to this disc at
every point. Then the map G∗(Z) associates to every point of ID a vector transversal to
f(ID) and so defines the desired extension X̃n of the vector field Xn. This proves the lemma.
Now by the Nijenhuis - Woolf theorem there exists a map h̃ζ : ID −→ M which is J-
holomorphic on ID such that h̃ζ = hζ for every ζ in a neighborhood of bID and the vector X̃
is tangent to hζ at the origin. Thus we can extend H to a function defined on ID × ID such
that the map H(ζ, •) is J-holomorphic for any ζ ∈ ID. This map H is a local diffeomorphism
and so determines an admissible parametrization of a neighborhood of f(ID) such that the
generated torus coincides with Λ. Theorem 5.6 now follows by Theorem 5.1.
6 Pushing discs through non-critical levels
In this section we explain how to push a given disc through non-critical level sets of a strictly
plurisubharmonic function.
Proposition 6.1 Suppose that ρ does not have critical values in the closed interval [c1, c2].
Let f : ID −→ Ωc1 be an immersed J-holomorphic disc such that f(bID) ⊂ bΩc1 . Then there
exists an immersed J-holomorphic disc f̃ : ID −→ Ωc2 such that f̃(0) = f(0), df̃(0) = λdf(0)
for some λ > 0 and f̃(bID) ⊂ bΩc2.
For the proof we need some preparations. Let ρ be a strictly plurisubharmonic function
on an almost complex manifold (M,J). For real c consider the domain Ωc = {ρ < c}.
Suppose that its boundary has no critical points. Let f : ID −→ Ωc be a J-holomorphic
disc of class C∞(ID) and such that f(bID) ⊂ bΩc. For every point p ∈ f(bID) consider a
J-holomorphic disc hp : 2ID −→ M touching bΩc from outside such that ρ ◦ hp|2ID\{0} > c.
We call the discs hp the Levi discs. The map hp can be chosen smoothly depending on
p ∈ f(bID).
An explicit construction of the Levi discs is given in [12]. In the almost complex case the
proof is similar; the only thing which has to be justified is the existence of discs hp touching
a strictly pseudoconvex level set from outside. This was recently proved by Barraud and
Mazzilli [1] and Ivashkovich and Rosay [16]. In [6] the result is obtained in any dimension.
For reader’s convenience we include a simple proof (see [6]).
Lemma 6.2 For a point p ∈ bΩc there exists a J-holomorphic disc hp such that hp(0) = p
and hp(ID\{0}) is contained in M\Ωc.
Proof : We fix local coordinates z = (z1, z2) near p such that p = 0 and J(0) = Jst.
Denote by ej , j = 1, 2 the vectors of the standard basis of IC
2. By an additional change of
coordinates we may achieve that the map h : ζ 7→ ζe1 is J-holomorphic on ID. We can
assume that the Levi form LJr (0, e1) = 1 so that
r(z) = 2Re z2 + 2Re
ajkzjzk +
αjkzjzk + o(|z|
α11 = ∆(r ◦ h)(0) = 1.
Now for every δ > 0 consider the non-isotropic dilation Λδ : (z1, z2) 7→ (δ
−1/2z1, δ
−1z2). The
J-holomorphicity of the map h implies that the direct images Jδ := (Λδ)∗(J) converge to Jst
as δ −→ 0 in the Ck norm for every positive integer k on any compact subset of IC2. Similarly,
the functions rδ := δ
−1r ◦ Λ−1 converge to the function r0 := 2Re z2 + |z1|
2 + 2Re βz21 (for
some β ∈ IC).
Consider a Jst-holomorphic disc ĥ : ζ 7→ ζe1 − βζ
2e2. According to the Nijenhuis-
Woolf theorem for every δ ≥ 0 small enough there exists a Jδ-holomorphic discs h
δ such
that the family (hδ)δ≥0 depends smoothly on the parameter δ and for every δ ≥ 0 we have
hδ(ζ) = ζe1 + o(|ζ |) and h
0 = ĥ. Since (r0 ◦ h
0)(ζ) = |ζ |2, we obtain that for δ > 0 small
enough that (rδ ◦ h
δ)(ζ) = Aδ(ζ) + o(|ζ |
2), where Aδ is a positive definite quadratic form on
IR2. Since the structures Jδ and J are biholomorphic, then the lemma follows.
Thus we obtain a smooth map
H : bID× ID −→M, H : (ζ, τ) 7→ hf(ζ)(τ) =: hζ(τ).
For simplicity we assume here that H is a local diffeomorphism although the Levi discs hζ
can intersect even for close values of ζ . We prove in a forthcoming paper that the pullback
H∗(J) of J to the bidisc can be defined even if H is not a local diffeomorphism. Thus
Λ := H(bID × bID) is an admissible torus and ρ|Λ ≥ c + ε for some ε > 0. We stress that ε
depends only on ρ (more precisely on a constant separating the norm of the gradient of ρ
from zero) and the C2-norm of J .
Now Theorem 5.6 implies that there exists a disc f̃ with the same direction as f at the
center and with the boundary attached to a torus arbitrarily close to Λ. Now we cut off the
discs hζ by the level set {ρ = c+ε/2} and obtain a disc with boundary attached to this level
set. Indeed, we have the following
Lemma 6.3 Suppose that ρ ◦ f |bID ≥ c0 and c0 is a non-critical value of ρ. Then there
exists a J-holomorphic disc f̃ centered at f(0) and tangent to f at the center with boundary
attached to the level set {ρ = c0}.
Proof : By the Hopf lemma the disc f intersects the level set {ρ = c0} transversally at
every point. Therefore the open set Ω = {ζ ∈ ID : ρ ◦ f(ζ) < c0} has a smooth bound-
ary. The set Ω may be disconnected, but the connected component of 0 ∈ Ω is simply
connected by the maximum principle applied to the function ρ ◦ f . Now the lemma follows
via reparametrization by the Riemann mapping theorem.
Then we again consider the Levi discs for this level set etc. By iterating this argument
a finite number of times we obtain Proposition 6.1.
7 Pushing discs through a critical level
In order to push the boundary of the disc f through critical level sets of ρ, we use a method
of [11, 7], which consists of temporarily switching to another plurisubharmonic function at
each critical level set. We need a version of the Morse lemma for almost complex manifolds.
Proposition 7.1 Let (M,J) be an almost complex manifold of complex dimension 2. Let
ρ be a strictly plurisubharmonic Morse function on M . Then there exists another strictly
plurisubharmonic Morse function ρ̃ close to ρ with the same critical points, such that at each
critical point of Morse index k in local coordinates given by Lemma 2.2 one has
ρ̃(z) = ρ̃(0) + |z1|
2 + |z2|
2 − a1Re z
1 − a2Re z
2 (17)
where
(i) a1 = a2 = 0 if k = 0,
(ii) a1 = 2 and a2 = 0 if k = 1,
(iii) a1 = a2 = 2 if k = 2.
Remark. This is a weak version of the Morse lemma because we change the given
function ρ instead of reducing it to a normal form.
The following result must be well known. For convenience we include a proof.
Lemma 7.2 Let B be a complex symmetric n×n matrix. Then there exists a unitary matrix
U such that U tBU is diagonal with nonnegative elements.
Proof : Using coordinate-free language, given a hermitian positive definite form H and a
complex symmetric bilinear form B on a vector space V , dimIC V = n, we need u1, ..., un ∈ V
such that
H(ui, uj) = δij , B(ui, uj) = ciδij, ci ≥ 0.
If the above holds with just ci ∈ IC, then by rotation ui 7→ σiui, |σi| = 1, we obtain ci ≥ 0.
It suffices to find u1 ∈ V , H(u1, u1) = 1, such that for every x ∈ V ,
H(x, u1) = 0 implies B(x, u1) = 0. (18)
Then the rest of ui in the H-orthogonal complement of u1 are found by induction. Given
u ∈ V , by duality, there is a unique vector L(u) ∈ V such that for every x ∈ V ,
H(x, L(u)) = B(x, u). (19)
Then L : V → V is a IR-linear (IC-antilinear) transformation. Since B is symmetric, then by
(19), L is real symmetric (self-adjoint) with respect to the form ReH . Then the eigenvalues
of L are real and the eigenvectors are in V (generally they are in V ⊗IR IC). Let u1 ∈ V
be an eigenvector of L, that is L(u1) = λu1, for some λ ∈ IR. We normalize u1 so that
H(u1, u1) = 1. Then for u = u1, (19) implies (18), and the lemma follows.
Proof of Proposition 7.1 : Let p be a critical point of ρ. Introduce a coordinate system
with the origin at p given by Lemma 2.2. In these coordinates the function ρ is strictly
plurisubharmonic at the origin with respect to Jst. Then
ρ(z) = ρ(0) +
aijzizj + Re
bijzizj +O(|z|
where aij = aji and bij = bji. By a linear transformation we can reduce to the form aij = δij .
If we now make a unitary transformation z 7→ Uz preserving |z1|
2 + |z2|
2, then the matrix
B = (bij) changes to U
tBU . By Lemma 7.2 the expression of ρ reduces to
ρ(z) = ρ(0) + |z1|
2 + |z2|
2 − Re (a1z
1 + a2z
2) +O(|z|
where aj ≥ 0, j = 1, 2. The remainder ϕ = O(|z|
3) can be removed by changing ρ to
ρ̃ = ρ−ϕλ, where λ(z) = λ0(z/ε) is a smooth cut-off function with λ0 ≡ 1 in a neighborhood
of the origin and λ0(z) = 0 for |z| ≥ 1, ε > 0 small enough.
Since ϕ(z) = O(|z|3), then |d(ϕλ)| ≤ C|z|2, ‖ ϕλ ‖C2(IC2)≤ Cε where C > 0 is independent
of ε. Since |dρ| ≥ C|z| in a neighborhood of 0 for some C > 0, then for small ε > 0 the
function ρ̃ has only one critical point at the origin, is strictly plurisubharmonic and matches
with ρ for |z| > ε.
The coefficients aj can be reduced to the standard values 0 and 2 depending on the index
k of the critical point. We need a cut-off function that falls down from 1 to 0 sufficiently
slowly.
Lemma 7.3 Given δ > 0 there exists a smooth non-increasing function φ with a compact
support on IR+ such that
(i) φ = 1 near the origin.
(ii) |tφ′(t)| ≤ δ.
(iii) |t2φ′′(t)| ≤ δ
The lemma follows because
Let bj = 0 (resp. 2) if 0 ≤ aj < 1 (resp. aj > 1). Let λ(z) = φ(|z|/ε), where φ is provided
by Lemma 7.3 for sufficiently small δ. Then the function
ρ̃(z) = ρ(z) + λ[(a1 − b1)Re z
1 + (a2 − b2)Re z
for sufficiently small ε has all the desired properties. Proposition 7.1 is proved.
Thus in what follows we assume that ρ has the properties given by Proposition 7.1. Let p
be a critical point of ρ and ρ(p) = 0. Without loss of generality assume that the index k of p
is equal to 1 or 2 since the disc obtained by Proposition 6.1 cannot approach a minimum of
ρ. Choose a small neighborhood U of p. By (17) ρ is strictly plurisubharmonic with respect
to Jst.
We apply the construction of Lemma 6.7 of [11]. Consider c0 > 0 small enough such that
0 is the only critical value of ρ in the interval [−c0, 3c0]. We can assume that c0 is small
enough so that the set K(c0) := {z : ρ(z) ≤ 3c0, |x
′|2 ≤ c0} is compactly contained in a
neighborhood of the origin corresponding to U . Here we use the notation x′ = x1, x
′′ = x2
and |x′|2 = x21 (resp. x
′ = (x1, x2) and |x
′|2 = x21 + x
2 ) if k = 1 (resp. k = 2). We will use
similar notations for the coordinates x, y and the coordinates u, v introduced below. Let
E = {y′ = 0, z′′ = 0, |x′|2 ≤ c0}. (20)
Then E is a totally real submanifold with boundary and dimE = k. Consider the isotropic
dilations of coordinates
dc0 : z 7→ w = u+ iv = c
Set Jc0 = (dc0)∗(J). The structures Jc0 converge to Jst in any C
m norm on compact subsets
of IC2 as c0 −→ 0. Consider the function ρ̂(w) := c
0 ρ(c
0 w). This function has no critical
values in [−1, 3] and its expression in the coordinates w = u+iv is the same as the expression
(17) of ρ that is
ρ̂(w) = 3v21 + v
2 − u
1 + u
if k = 1 and
ρ̂(w) = 3v21 + 3v
2 − u
1 − u
if k = 2. In particular the set K = dc0(K(c0)) is given by {w : ρ̂(w) ≤ 3, |u
′|2 ≤ 1} and is a
fixed compact independent of c0.
It is important that the origin is a critical point of the function ρ and the local coordinates
and the function ρ are given by Proposition 7.1. This allows to use the isotropic dilations in
contrast with Lemma 6.2.
Since the function ρ̂ is strictly plurisubharmonic with respect to Jst, we can apply the
construction of [11] (Lemma 6.7 and section 6.4). We replace the function ρ̂ by a new
function ϕ defined by
ϕ(w) = 3v21 + v
2 − h(u
1) + u
if k = 1 and
ϕ(w) = 3v21 + 3v
2 − h(u
1 + u
if k = 2, where h ≥ 0 is a suitable function. The construction of h depends on the parameter
c0 only. In our “delated” coordinates w we apply this construction taking c0 = 1. Namely,
according to [11] there exist constants 0 < τ0 < τ1 < 1 depending on the eigenvalues of ρ̂
and a function ϕ strictly plurisubharmonic on IC2 with respect to Jst satisfying the following
properties:
(i) ρ̂ ≤ ϕ ≤ ρ̂+ τ1,
(ii) ρ̂+ τ0 ≤ ϕ on the set {|u
′|2 ≥ τ0}
(iii) ϕ = ρ̂+ τ1 on {|u
′|2 ≥ 1}
Since ρ̂ is strictly plurisubharmonic with respect to the structure Jc0 , the function ϕ also is
strictly Jc0-plurisubharmonic on {|u
′|2 ≥ 1} in view of (iii). On the other hand the structures
Jc0 converge to Jst in any C
m norm on compact subsets of IC2 as c0 −→ 0. Therefore, since
ϕ is strictly Jst-plurisubharmonic, it also is strictly Jc0-plurisubharmonic on K if c0 is small
enough. Thus, ϕ is strictly Jc0-plurisubharmonic on {ρ̂ ≤ 3}.
Now consider the function ρ̃(z) = c0ϕ(c
0 z) and set t0 = τ0c0.
The function ρ̃ satisfies the following properties:
(i) ρ̃ is strictly plurisubharmonic (with respect to Jst) in a neighborhood V ⊂ U of 0 and
ρ̃ = ρ+ t1 on the complement of V . Here t1 > 0 is a constant.
(ii) ρ̃ has no critical values on (0, 3c0)
(iii) There exists t0 ∈ (0, c0) such that
{ρ ≤ −c0} ∪ E ⊂ {ρ̃ ≤ 0} ⊂ {ρ ≤ −t0} ∪ E, (21)
where E is defined above by (20).
(iv) We have
{ρ ≤ c0} ⊂ {ρ̃ ≤ 2c0} ⊂ {ρ < 3c0} (22)
By Proposition 6.1 we construct an immersed J-holomorphic disc f such that −t0 <
ρ ◦ f |bID < 0. The boundary of f is contained in a torus Λ formed by discs complex tangent
to a level set of ρ. We will perturb the disc f slightly in order to avoid the intersection of
its boundary with E.
Proposition 7.4 Let f : ID −→ M be an immersed J-holomorphic disc in (M,J), where
dimICM = 2. Let E be a smooth submanifold in M . Then for every m ≥ 2 there exists a
J-holomorphic disc f̃ arbitrarily close to f in Cm(ID) such that f̃(0) = f(0), df̃(0) = df(0),
and f̃ |bID is transverse to E. In particular, if dimIR E ≤ 2, then f̃(bID) ∩ E = ∅.
Proof : By the implicit function theorem, the restriction f |bD admits infinitesimal pertur-
bations in all directions. Then the proposition follows by the proof of Thom’s transversality
theorem.
We now assume f(bID) ∩ E = ∅. In view of the inclusion (21) we conclude that ρ̃ > 0
on f(bID). By Lemma 6.3 we cut off the disc f by a level set {ρ̃ = c} for some c > 0 to
assume that now f(bID) is contained in this level set. The function ρ̃ has no critical values
in (0, 3c0). By Proposition 6.1 applied to the disc f and the function ρ̃ there exists a new
disc f̃ with the boundary contained in {ρ̃ > 2c0}. In view of (22) we have the inclusion
{ρ̃ > 2c0} ⊂ {ρ > c0}. Now the boundary of f̃ is outside the critical level {ρ = 0} as desired,
and we switch back to the original function ρ.
8 Proof of Theorem 1.1
Since the function ρ is strictly plurisubharmonic, then after a generic perturbation of ρ which
does not change the given level set, we can assume that ρ is a Morse function. Let p be the
given point in D. If p is not a point of minimum of ρ, we proceed as follows. Consider a
small J-holomorphic disc f centered at p with the given direction v. Consider a non-critical
level set ρ = c such that ρ(p) < c. Consider a foliation of a neighborhood of f by a complex
one-parameter family of J-holomorphic discs hq, q ∈ f(ID) such that the boundaries of these
discs are outside the sublevel set ρ < c. When q runs over the circle f(bID) these boundaries
form a torus. Applying Proposition 5.1 we obtain a new disc f̃ centered at p and still in the
same direction at p but with ρ ◦ f̃ |bID > 0.
If p is a point of minimum for ρ, we drop this first step and directly have this situation
with f̃ = f . Now the desired results follow by Proposition 6.1 combined with the above
argument allowing to push boundaries of discs through critical levels.
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  We prove the existence of global Bishop discs in a strictly pseudoconvex
Stein domain in an almost complex manifold of complex dimension 2.

Introduction
The problem of embedding complex discs or general Riemann surfaces into complex manifolds
has been well-known for a long time. The interest to the case of almost complex manifolds
has grown due to a strong link with symplectic geometry (Gromov [13]). We present the
following result.
Theorem 1.1 Let (M,J) be an almost complex manifold of complex dimension 2 admitting
a strictly plurisubharmonic exhaustion function ρ. Then for every non-critical value c of ρ,
every point p ∈ Ωc = {ρ < c} and every vector v ∈ Tp(M) there exists a J-holomorphic
immersion f : ID −→ Ωc, where ID ⊂ IC is the unit disc, such that f(bID) ⊂ bΩc, f(0) = p,
and df0
∂Re ζ
= λv for some λ > 0.
For a domainM ⊂ ICn with the standard complex structure, the result is due to Forstnerič
and Globevnik [12]; there are various generalizations including embedding bordered Riemann
surfaces into singular complex spaces (see [7] and references there).
http://arxiv.org/abs/0704.0124v3
Recently Biolley [4] proved a similar result for an almost complex manifold M of any
dimension n, but under the additional hypothesis that the defining function ρ is subcritical.
The latter means that ρ does not have critical points of the maximum Morse index n. (A
plurisubharmonic function can not have critical points of index higher than n.) We don’t
impose such a restriction. Furthemore, Biolley [4] does not prescribe the direction of the disc.
Her method is based on the Floer homology and substantially uses recent work of Viterbo
[23] and Hermann [14]. Our proof is self-contained; we adapt the ideas of Forstnerič and
Globevnik [12] to the almost complex case using the methods of classical complex analysis
and PDE.
In most work on the existence of global discs with boundaries in prescribed totally real
manifolds ([2, 9, 10, 15, 17] and others) the authors use the continuity principle. By the
implicit function theorem and the linearized equation they show that any given disc generates
a family of nearby discs. Then the compactness argument allows for passing to the limit. In
contrast, we construct the discs by solving the almost Cauchy-Riemann equation directly.
Following [12], we start with a small disc passing through the given point in given direction
and push the boundary of the disc in the directions complex-tangent to the level sets of the
defining function ρ; it results in increasing ρ due to pseudoconvexity. This plan leads to
a problem of attaching J-holomorphic discs to totally real tori in a level set of ρ. The
problem is of independent interest and may occur elsewhere. It reduces in turn to the
existence theorem for a boundary value problem for a quasilinear elliptic system of partial
differential equations in the unit disc (Theorem 4.1). We prove it by the classical methods of
the Beltrami equations and quasiconformal mappings (Ahlfors, Bers, Boyarskii, Lavrentiev,
Morrey, Vekua; see [3, 21] and references there). The result can be viewed as a far reaching
generalization of the Riemann mapping theorem.
Since the almost Cauchy-Riemann equation is nonlinear, one can only hope to find a
solution close to a current disc f . By measuring the closeness in the Lp norm, we are able in
fact to construct a disc sufficiently far from f in the sup-norm. To make sure we are looking
for a disc close to f , we adapt the idea of [12] of adding to f(ζ) a term with a factor of ζn
(ζ ∈ ID) with big n. We develop a nonlinear version of this idea.
The above procedure works well in the absence of critical points of ρ. In order to push
the boundary of the disc through critical level sets, we use a method by Drinovec Drnovšek
and Forstnerič [7, 11], which consists of temporarily switching to another plurisubharmonic
function at each critical level set. We point out that adapting this method to the almost
complex case is not a major problem because the difficulties are localized near the critical
points, in which the almost complex structure can be closely approximated by the standard
complex structure.
Although higher dimension gives one more freedom for constructing J-holomorphic discs,
we must admit that our proof of the main result goes through in dimension 2 only. The reason
is that our main tool (Theorem 4.1) needs a special coordinate system in which coordinate
hyperplanes z = const are J-complex, which generally can be achieved only in dimension
2. For a domain in ICn with the standard complex structure, the result is obtained in [12]
by reduction to dimension 2 using sections by 2-dimensional complex hypersurfaces. Such a
reduction in not possible for almost complex structures.
We thank Franc Forstnerič and Josip Globevnik for helpful discussions, in particular, for
pointing out at some difficulties in the problem and for the important references [7, 11].
Parts of the work were completed when the third author was visiting Université de
Provence and Université des Sciences et Technologies de Lille in the spring of 2006. He
thanks these universities for support and hospitality.
2 Almost complex manifolds
Let (M,J) be an almost complex manifold. Denote by ID the unit disc in IC and by Jst the
standard complex structure of ICn; the value of n is usually clear from the context. Let f be a
smooth map from ID intoM . Recall that f is called J-holomorphic if df ◦Jst = J◦df . We also
call such a map f a J-holomorphic disc or a pseudoholomorphic disc or just a holomorphic
disc when a complex structure is fixed. We will often denote by ζ the standard complex
coordinate on IC.
A fundamental result of the analysis and geometry of almost complex structures is the
Nijehnuis–Woolf theorem which states that given point p ∈ M and given tangent vector
v ∈ TpM there exists a J-holomorphic disc f : ID −→M centered at p, that is, f(0) = p and
such that df(0)(∂/∂Re ζ) = λv for some λ > 0. This disc f depends smoothly on the initial
data (p, v) and the structure J . A short proof of this theorem is given in [19]. This result
will be used several times in the present paper.
It is well known that an almost complex manifold (M,J) of complex dimension n can be
locally viewed as the unit ball IB in ICn equipped with an almost complex structure which
is a small deformation of Jst. More precisely, let (M,J) be an almost complex manifold of
complex dimension n. Then for every p ∈M , δ0 > 0, and k ≥ 0 there exist a neighborhood U
of p and a smooth coordinate chart z : U −→ IB such that z(p) = 0, dz(p) ◦ J(p) ◦ dz−1(0) =
Jst, and the direct image z∗(J) := dz ◦ J ◦ dz
−1 satisfies the inequality ||z∗(J)− Jst||Ck(ĪB) ≤
δ0. For a proof we point out that there exists a diffeomorphism z from a neighborhood
U ′ of p ∈ M onto IB such that z(p) = 0 and dz(p) ◦ J(p) ◦ dz−1(0) = Jst. For δ > 0
consider the isotropic dilation dδ : t 7→ δ
−1t in ICn and the composite zδ = dδ ◦ z. Then
limδ→0 ||(zδ)∗(J)−Jst||Ck(ĪB) = 0. Setting U = z
δ (IB) for positive δ small enough, we obtain
the desired result. As a consequence we obtain that for every point p ∈ M there exists a
neighborhood U of p and a diffeomorphism z : U → IB with center at p (in the sense that
z(p) = 0) such that the function |z|2 is J-plurisubharmonic on U and z∗(J) = Jst +O(|z|).
Let u be a function of class C2 on M , let p ∈ M and v ∈ TpM . The Levi form of u at p
evaluated on v is defined by LJ(u)(p)(v) := −d(J∗du)(v, Jv)(p).
The following result is well known (see, for instance, [6]).
Proposition 2.1 Let u be a real function of class C2 on M , let p ∈M and v ∈ TpM . Then
LJ(u)(p)(v) = ∆(u◦f)(0) where f : rID −→M for some r > 0 is an arbitrary J-holomorphic
map such that f(0) = p and df(0)(∂/∂Re ζ) = v, ζ ∈ rID.
The Levi form is invariant with respect to J-biholomorphisms. More precisely, let u be a
C2 real function onM , let p ∈M and v ∈ TpM . If Φ is a (J, J
′)-holomorphic diffeomorphism
from (M,J) into (M ′, J ′), then LJ(u)(p)(v) = LJ
(u ◦ Φ−1)(Φ(p))(dΦ(p)(v)).
Finally, it follows from Proposition 2.1 that a C2 function u is J-plurisubharmonic on
M if and only if LJ (u)(p)(v) ≥ 0 for all p ∈ M , v ∈ TpM . Thus, similarly to the case of
the integrable structure one arrives in a natural way to the following definition: a C2 real
valued function u on M is strictly J-plurisubharmonic on M if LJ(u)(p)(v) is positive for
every p ∈M , v ∈ TpM\{0}.
Let J be a smooth almost complex structure on a neighborhood of the origin in ICn and
J(0) = Jst. Denote by z = (z1, ..., zn) the standard coordinates in IC
n (in matrix computations
below we view z as a column). Then a map z : ID −→ ICn is J-holomorphic if and only if it
satisfies the following system of partial differential equations
zζ − A(z)zζ = 0, (1)
where A(z) is the complex n× n matrix defined by
A(z)v = (Jst + J(z))
−1(Jst − J(z))v (2)
It is easy to see that right-hand side of (2) is IC-linear in v ∈ ICn with respect to the standard
structure Jst, hence A(z) is well defined. Since J(0) = Jst, we have A(0) = 0. Then in a
sufficiently small neighborhood U of the origin the norm ‖ A ‖L∞(U) is also small, which
implies the ellipticity of the system (1).
However, we will need a more precise choice of coordinates imposing additional restric-
tions on the matrix function A. The proof of the following elementary statement can be
found, for instance, in [6].
Lemma 2.2 After a suitable polynomial second degree change of local coordinates near the
origin
z 7→ z +
akjzkzj
we can achieve
A(0) = 0, Az(0) = 0
In these coordinates the Levi form of a given C2 function u with respect to J at the origin
coincides with its Levi form with respect to Jst that is
LJ(u)(0)(v) = LJst(u)(0)(v)
for every vector v ∈ T0IR
3 Integral transforms in the unit disc
Let Ω be a domain in IC. Let TΩ denote the Cauchy-Green transform
TΩf(ζ) =
f(τ)dτ ∧ dτ
τ − ζ
. (3)
Let RΩ denote the Ahlfors-Beurling transform
RΩf(ζ) =
f(τ)dτ ∧ dτ
(τ − ζ)2
, (4)
where the integral is considered in the sense of the Cauchy principal value. We omit the
index Ω if it is clear form the context. Denote by B the Bergman projection for ID.
Bf(ζ) =
f(τ)dτ ∧ dτ
(τζ − 1)2
We need the following properties of the above operators.
Proposition 3.1 (i) Let p > 2 and α = (p−2)/p. Then the linear operator T : Lp(ID) −→
Cα(IC) is bounded, in particular, T : Lp(ID) −→ L∞(ID) is compact. If f ∈ Lp(ID),
then ∂ζTf = f , ζ ∈ ID, as a Sobolev derivative.
(ii) Let m ≥ 0 be integer and let 0 < α < 1. Then the linear operators T : Cm,α(ID) −→
Cm+1,α(IC) and R : Cm,α(ID) −→ Cm,α(ID) are bounded. Furthermore,if f ∈ Cm,α(ID),
then ∂ζTf = f and ∂ζTf = Rf , ζ ∈ ID, in the usual sense.
(iii) The operator RΩ can be uniquely extended to a bounded linear operator RΩ : L
p(Ω) −→
Lp(Ω) for every p > 1. If f ∈ Lp(ID), p > 1 then ∂ζTf = Rf as a Sobolev derivative.
Moreover, the operator RIC is an isometry of L
2(IC), therefore ‖ RIC ‖L2(IC)= 1.
(iv) The Bergman projection B : Lp(ID) −→ Ap(ID) is bounded. Here Ap(ID) denotes the
space of all holomorphic functions in ID of class Lp(ID).
(v) The functions p 7→‖ T ‖Lp(Ω) and p 7→‖ R ‖Lp(Ω) are logarithmically convex and in
particular, continuous for p > 1.
The proofs of the parts (i)–(iii) are contained in [21]. The part (iv) follows from (iii); see e.
g. [8]. The part (v) follows by the classical interpolation theorem of M. Riesz–Torin (see e.
g. [24]).
We introduce modifications of the operators T and R for solving certain boundary value
problems in the unit disc ID. For f ∈ Lp(ID) we define
T0f(ζ) = Tf(ζ)− Tf(ζ
), ζ ∈ ID. (5)
By Proposition 3.1 for p > 2 and α = (p− 2)/p, the linear operator T0 : L
p(ID) −→ Cα(ID)
is bounded, in particular, T0 : L
p(ID) −→ L∞(ID) is compact. Since the function Tf is
holomorphic and bounded in IC\ID, then the function ζ 7→ (Tf)(ζ
) is holomorphic in ID.
Hence ∂ζT0f = ∂ζTf = f . Furthermore, for ζ ∈ bID, we have ζ = ζ
, therefore by (5),
ReT0f(ζ) = 0. Hence for f ∈ L
p(ID), the function u = T0f solves the boundary value
problem
∂ζu = f, ζ ∈ ID,
Reu|bID = 0
We further define
R0f := ∂ζT0f.
Since ∂ζTf = Rf and ∂ζTf = f , then
R0f(ζ) = ∂ζT0f(ζ) = Rf(ζ)− ∂ζTf(ζ
) = Rf(ζ) + ζ−2Rf(ζ
), (6)
and we obtain a nice formula
R0f = Rf +Bf,
where B is the Bergman projection. By Propositions 3.1(iv) and (v), the operator R0 :
Lp(ID) −→ Lp(ID) is bounded, and the map p 7→‖ R0 ‖Lp(ID) is continuous for p > 1. By
Proposition 3.1(iii), R is an isometry of L2(IC). The analogue of this result for the operator
R0 may have been used for the first time by Vinogradov [22]. In fact we came across [22]
after proving the following
Theorem 3.2 R0 is a IR-linear isometry of L
2(ID), in particular, ‖ R0 ‖L2(ID)= 1.
Since we could not find a proof in the literature, for completeness we include it here.
Proof : For a domain G ⊂ IC we use the inner product
(f, g)G = −
fgdζ ∧ dζ.
We put
σf(ζ) = ζ
), ψ(ζ) = ζ
Then σ2 = id. By substitution ζ 7→ ζ
we obtain
(σf, σg)ID = (g, f)IC\ID, Rσ = ψσR, R = ψσRσ. (7)
By (6) we have
R0f = Rf + ψσRf.
Let f ∈ L2(ID). Extend f to all of IC by putting f(ζ) = 0 for |ζ | > 1. Then
‖ R0f ‖
L2(ID)= (Rf + ψσRf,Rf + ψσRf)ID =
(Rf,Rf)ID + 2Re (Rf, ψσRf)ID + (ψσRf, ψσRf)ID.
Since |ψ| = 1, by (7) we obtain
(ψσRf, ψσRf)ID = (σRf, σRf)ID = (Rf,Rf)IC\ID,
(Rf, ψσRf)ID = (ψσRσf, ψσRf)ID = (Rσf,Rf)IC\ID = (ψσRf,Rf)IC\ID = (Rf, ψσRf)IC\ID.
Then by the previous line and because R is an isometry
2Re (Rf, ψσRf)ID = Re (Rf, ψσRf)IC = Re (Rf,Rσf)IC = Re (f, σf)IC = 0.
Hence
‖ R0f ‖
L2(ID)= (Rf,Rf)ID + (Rf,Rf)IC\ID =‖ Rf ‖
L2(IC)=‖ f ‖
L2(IC)=‖ f ‖
L2(ID),
which proves the theorem.
4 Riemann mapping theorem for an elliptic system
The Riemann mapping theorem asserts that for every simply connected domain G ⊂ IC there
exists a conformal map of G onto ID. If G is smooth, then there is a diffeomorphism f :
G −→ ID, which defines an almost complex structure J = f∗(Jst) in ID. Then the Riemann
mapping theorem reduces to constructing a J-holomorphic map z : (ID, Jst) −→ (ID, J). The
latter satisfies the Beltrami type equation ∂ζz = A(z)∂ζz, which is equivalent to the linear
Beltrami equaion ∂zζ + A(z)∂zζ = 0. We consider the following more general system
∂ζz = a(z, w)∂ζz,
∂ζw = b(z, w)∂ζz,
which cannot be reduced to a linear one. Here z, w are unknown functions of ζ ∈ ID and a, b
are C∞ coefficients. By eliminating ζ , the system reduces to a nonhomogeneous quasilinear
Beltrami type equation ∂zw + a∂zw = b, but we prefer to deal with (8) directly.
The following theorem is our main technical tool for constructing pseudoholomorphic
discs with boundaries in a prescribed torus. For r > 0 denote IDr := rID.
Theorem 4.1 Let a, b : ID× ID1+γ −→ IC (γ > 0) be smooth functions such that
a(z, 0) = b(z, 0) = 0 and |a(z, w)| ≤ a0 < 1.
Then there exists C > 0 such that for every integer n ≥ 1 the system (8) admits a smooth
solution (zn, wn) with the following properties:
(i) |zn(ζ)| = |wn(ζ)| = 1 for |ζ | = 1.
(ii) zn : ID −→ ID is a diffeomorphism with zn(0) = 0.
(iii) |wn(ζ)| ≤ C|ζ |
n, |wn(ζ)| < 1 + γ.
Proof : Shrinking γ > 0 if necessary, we extend the functions a and b to all of IC2
preserving their properties. We will look for a solution of (8) in the form
z = ζeu, w = ζnev.
Then for the new unknowns u and v we have the following boundary value problem
∂ζu = A(u, v, ζ)(1 + ζ∂ζu), ζ ∈ ID
∂ζv = B(u, v, ζ)(1 + ζ∂ζu), ζ ∈ ID
Re u(ζ) = Re v(ζ) = 0, |ζ | = 1
where
A = aζ−1eu−u,
B = bζ−neu−v.
Put ∂ζu = h and choose u in the form u = T0h. Then ∂ζu = R0h, which we plug into (9).
We obtain the following system of singular integral equations for u, v and h:
h = A(1 + ζR0h),
u = T0h,
v = T0(B(1 + ζR0h))
We denote by ‖ f ‖p the L
p-norm of f in ID. Since the function p 7→‖ R0 ‖p is continuous in
p and ‖ R0 ‖2= 1 we choose p > 2 such that
a0 ‖ R0 ‖p< 1.
For given u, v ∈ L∞(ID) the map h 7→ A(1 + ζR0h) is a contraction in L
p(ID) because
‖ ζA ‖∞‖ R0 ‖p< 1.
Hence there exists a unique solution h = h(u, v) of the first equation of (10) satisfying
‖ h ‖p≤
‖ A ‖p
1− a0 ‖ R0 ‖p
Consider the map F : L∞(ID)× L∞(ID) −→ L∞(ID)× L∞(ID) defined by
F : (u, v) 7→ (U, V ) = (T0h, T0(B(1 + ζR0h)))
where h = h(u, v) is determined above. Then F is continuous (even Lipschitz) map. Let
E = {(u, v) ∈ L∞(ID)× L∞(ID) :‖ u ‖∞≤ u0, ‖ v ‖∞≤ v0}
We need the following
Lemma 4.2 There exist u0 > 0, v0 > 0 such that E is invariant under F .
Assuming the lemma, we prove the existence of the solution of (10). Indeed, since
T0 : L
p(ID) −→ L∞(ID) is compact for p > 2, then F : E −→ E is compact. Since
E is a bounded, closed and convex, then the existence of the solution of (10) follows by
Schauder’s principle.
Proof of Lemma 4.2 : Since a(z, 0) = b(z, 0) = 0, we have
|a(z, w)| ≤ C1|w|, |b(z, w)| ≤ C1|w|.
Here and below we denote by Cj constants independent of n. We have
|a| = |a(ζeu, ζnev)| ≤ C1e
‖v‖∞ |ζ |n,
‖ A ‖p=‖ aζ
−1 ‖p≤ C2 ‖ ζ
n−1 ‖p e
‖v‖∞ ≤ C3e
‖v‖∞n−1/p.
By (11), ‖ h ‖p≤ C4e
‖v‖∞n−1/p, hence
‖ U ‖∞≤ C5e
‖v‖∞n−1/p.
Similarly
|B| = |b(ζeu, ζnev)ζ−neu−v| ≤ C1e
‖u‖∞ ,
‖ B ‖∞≤ C1e
‖ V ‖∞≤ C7(‖ B ‖p + ‖ B ‖∞‖ h ‖p) ≤ C8e
‖u‖∞ .
Let δ = n−1/p. Then
‖ U ‖∞≤ C9δe
‖v‖∞ ,
‖ V ‖∞≤ C9e
Consider the system
u0 = C9δe
v0 , v0 = C9e
with the unknowns u0, v0. Then
u0 = C9δe
For small δ > 0 this equation has two positive roots. Let u0 = u0(δ) be the smaller root and
v0 = v0(δ) = C9e
u0. Now if ‖ u ‖∞≤ u0, ‖ v ‖∞≤ v0, then
‖ U ‖∞≤ C9δe
‖v‖∞ ≤ C9δe
v0 ≤ u0,
‖ V ‖∞≤ C9δe
‖u‖∞ ≤ C9δe
u0 ≤ v0
Hence E is invariant under F , which proves the lemma.
Thus the solution of (10) in L∞(ID) exists for n big enough. Since h ∈ Lp(ID), p > 2,
the second and the third equations of (10) imply that u, v ∈ Cα(ID), α = (p − 2)/p. Since
∂ζu = h ∈ L
p(ID) and ∂ζu = R0h ∈ L
p(ID) as Sobolev’s derivatives, then u and v are
solutions of (9), hence z = ζeu and w = ζnev are solutions of (8). By the ellipticity of
the system, z, w ∈ C∞(ID). The smoothness up to the boundary can be derived directly
from the properties of the Beltrami equation; it also follows by the reflection principle for
pseudoholomorphic discs attached to totally real manifolds (see, e.g., [18]).
Since the winding number of z|bID about 0 equals 1 and
∣∂ζz/∂ζz
∣ = |a| ≤ a0 < 1 then
z : ID −→ ID is a homeomorphism by the classical properties of the Beltrami equation [21],
and (ii) follows.
Note that u0 −→ 0, v0 −→ C9 as n −→ ∞. Since T0 : L
p(ID) −→ Cα(ID) is bounded,
then we have
‖ v ‖Cα(ID)≤ C10, ‖ e
v ‖Cα(ID)≤ C11,
and |w(ζ)| ≤ C11|ζ |
n. Furthermore, since |ev| = 1 on bID, then |ev(ζ)| ≤ 1 + C11(1 − |ζ |)
for |ζ | < 1. Then |w(ζ)| ≤ |ζ |n(1 + C11(1 − |ζ |)
α), hence ‖ w ‖∞−→ 1 as n −→ ∞. Hence
‖ w ‖∞< 1+ γ for n big enough, and (iii) follows. This completes the proof of Theorem 4.1.
5 Pseudoholomorphic discs attached to real tori
This section concerns the geometrization of Theorem 4.1. We apply Theorem 4.1 in order
to obtain a crucial technical result on (approximately) attaching pseudoholomorphic discs
to a given real 2-dimensional torus in (M,J). We will use this result later for pushing discs
across level sets of the defining function ρ in Theorem 1.1.
The tori and the discs considered in this section are not arbitrary. We study a special
case which will suffice for the proof of the main result. Given a psedoholomorphic immersed
disc f , we associate with f a real 2-dimensional torus Λ formed by the boundary circles of
discs hζ centered at the boundary points f(ζ), ζ ∈ bID. Thus, our initial data is a pair (f,Λ).
Our goal is to construct a pseudoholomorphic disc with the boundary attached to the torus
Λ. First we find a suitable neighborhood of the disc f which can be parametrized by the
bidisc in IC2. We transport the structure J onto this bidisc and choose the coordinates there
such that the equations for J-holomorphic discs take the form used in Theorem 4.1. The
theorem will provide a pseudoholomorphic disc approximately attached to Λ.
5.1 Admissible parametrizations by the bidisc and generated tori
Let f : ID −→ (M,J) be a J-holomorphic disc of class C∞(ID). Suppose f is an immersion.
Let γ > 0. Given ζ ∈ D consider a J-holomorphic disc
hζ : (1 + γ)ID −→ M
satisfying the condition hζ(0) = f(ζ) and such that the direction dhζ(0)(
∂Re τ
) is not tangent
to f . Admitting some abuse of notation, we sometimes write hf(ζ) for hζ .
This allows to define a C∞ map
H : ID× (1 + γ)ID −→M, H(ζ, τ) = hζ(τ).
Then H has the following properties:
(i) For every ζ ∈ ID the map hζ := H(ζ, •) is J-holomorphic.
(ii) For every ζ ∈ ID we have H(ζ, 0) = f(ζ).
(iii) For every ζ ∈ ID the disc hζ is transversal to f at the point f(ζ).
We assume in addition that
(iv) H : ID× (1 + γ)ID −→M is locally diffeomorphic.
Then Λ = H(bID×(1+γ)bID) is a real 2-dimensional torus immersed intoM . It is formed
by a family of topological circles γζ = hζ((1 + γ)bID) parametrized by ζ ∈ bID. Every such a
circle bounds a J-holomorphic disc hζ : (1 + γ)ID −→ M centered at f(ζ). In particular the
torus Λ can be continuously deformed to the circle f(bID).
If the above conditions (i) - (iv) hold we say that a mapH is an admissible parametrization
of a neighborhood of f(ID) and Λ is the torus generated by H .
5.2 Ellipticity of admissible parametrizations
We prove the following consequence of Theorem 4.1.
Theorem 5.1 Let f : ID −→ (M,J) be a C∞ immersion J-holomorphic in ID. Suppose
that there exists an admissible parametrization H of a neighborhood of f(ID) and let Λ be
the generated torus. Then there exists an immersed J-holomorphic disc f̃ of class C∞(ID)
centered at f(0), tangent to f at f(0) and satisfying the boundary condition f̃(bID) ⊂ H(bID×
bID).
We stress that the boundary of f̃ is attached to the torus H(bID×bID) and not to Λ. However
since γ > 0 can be chosen arbitrarily close to 0, this leads to the following result sufficient
for applications.
Corollary 5.2 In the hypothesis of the former theorem for any positive integer n there exists
an immersed J-holomorphic disc fn of class C∞(ID) centered at f(0), tangent to f at f(0)
and such that dist(fn(bID),Λ) −→ 0 as n −→ ∞.
Here dist denotes any distance compatible with the topology of M .
We begin the proof of Theorem 5.1 with the remark that the discs hζ , ζ ∈ D, fill a subset
V of M containing f(ID) which can be viewed as a fiber space with the base f(ID) and the
generic fiber hζ((1 + γ)ID). Therefore the defined above map
H : ID× (1 + γ)ID −→ V
gives a natural parametrization of V by the bidisc Uγ := ID × (1 + γ)ID. Since H is locally
diffeomorphic (see (iv) above) the inverse map H−1 is defined in a neighborhood of every
point of V . This allows to define the almost complex structure J̃ = H∗(J) = dH−1 ◦ J ◦ dH
on Uγ . The structure J̃ has a special form. Indeed, in the standard basis of IR
4 we have
J̃11 J̃12
J̃21 J̃22
where J̃kj are real 2×2 matrices. We recall that in this basis the standard complex structure
st of IC has the form
It follows by the property (i) of H that the maps τ 7→ (c, τ) are J̃-holomorphic for every
fixed c. This implies that J̃12 = 0 and J̃22 = J
st . Furthermore, since the map ζ 7→ (ζ, 0) is
J̃-holomorphic, we have J̃11(z, 0) = J
st and J̃21(z, 0) = 0.
Let now g : ID −→ Uγ be a J̃ -holomorphic map. If we set ζ = ξ+iη, the Cauchy–Riemann
equations have expressing the J̃-holomorphicity of g have the form
= 0 (13)
Suppose now that the matrix Jst+J is invertible. Then the Cauchy–Riemann equations can
be rewritten in the form
gζ + A(g)gζ = 0 (14)
where A is defined by (2). If we use the notation g = (z, w), then the Cauchy–Riemann
equations (14) can be written in the form
∂ζz = a(z, w)∂ζz,
∂ζw = b(z, w)∂ζz
identical to (8). Furthermore, since J̃(z, 0) = Jst, the conditions a(z, 0) = b(z, 0) = 0 are
satisfied.
Proposition 5.3 We have ‖ a ‖∞< 1.
Proof : The proof consists of two steps. First we study the matrix J̃+Jst which determines
the matrix A in the Cauchy–Riemann equations (14).
Lemma 5.4 The matrix J̃(z, w) + Jst is non-degenerate for any (z, w) ∈ ID× (1 + γ)ID.
Proof : It suffices to verify the condition det(J̃11(z, w) + J
st ) 6= 0. For every fixed (z, w)
the matrix J̃11(z, w) defines a complex structure on the euclidean space IR
2 so there exists a
matrix P = P (z, w) such that
J̃11(z, w) = PJ
−1. (16)
Recall that the manifold J2 of all complex structures on IR
2 can be identified with the
quotient GL(2, IR)/GL(1, IC) and has two connected components: J +2 and J
2 . A structure
J̃11 belongs to J
2 (resp. to J
2 ) if in the representation (16) we have detP > 0 (resp.
detP < 0). Suppose now that det(PJ
st ) = 0 or equivalently det(PJ
st +J
st P ) =
0 at some point (z, w). If we denote by pjk the entries of the matrix P , the last equality means
jk=1 p
jk = 0 which together with the non-degeneracy of P implies that detP < 0 so
that J̃11(z, w) ∈ J
2 . On the other hand, for the point (z, 0) we have detP > 0 since
J̃11(z, 0) = J
st so J̃(z, 0) ∈ J
2 . But we can join the points (z, 0) and (z, w) by a real
segment, so this contradiction proves lemma.
Now we can conclude the proof of Proposition 5.3. It follows by Lemma 5.4 that the
Cauchy–Riemann equations (13) can be written in the form (15) on ID × (1 + γ)ID. The
Cauchy–Riemann equations are elliptic at every point and this condition is independent of
the choice of the coordinates. The system (15) is ellipitic at a point (z, w) if and only if
|a(z, w)| 6= 1. Since a(z, 0) = 0 we obtain by connectedness that |a| < 1 on ID × (1 + γ)ID,
which concludes the proof.
Now Theorem 5.1 follows by Theorem 4.1.
5.3 Construction of an admissible parametrization with a pre-
scribed generated torus
So far we studied a situation where an admissible parametrization of a neighborhood of
an immersed J-holomorphic disc was given and proved the existence of discs with bound-
aries close to the generated torus. In the proof of our main result, we need an admissible
parametrization of a neighborhood of a J-holomorphic disc with a given generated torus.
Let f : ID −→ M be an immersed J-holomorphic disc of class C∞(ID). We extend f
smoothly to a neighborhood of ID. Let U be a small neighborhood of bID. For every point
f(ζ), ζ ∈ U , consider a J-holomorphic disc hζ : 2ID −→ M . Suppose that the map hζ
smoothly depends on ζ ∈ U . Thus we obtain a smooth map
H : bID × ID −→ M, H : (ζ, τ) 7→ hζ(τ).
Then Λ := H(bID × bID) is a real 2-dimensional torus. In order to construct an admissible
parametrization with the generated torus Λ we need to extend the map H from the cylinder
bID × ID to the bidisc ID× ID.
Definition 5.5 We call the described above torus Λ admissible. We further put Xζ :=
Xf(ζ) = dhζ(0)(
∂Re τ
) for every ζ ∈ U .
Theorem 5.6 Let f : ID −→ (M,J) be an immersed J-holomorphic disc of class C∞(ID).
Let Λ be an admissible torus. Then there is a sequence of admissible tori Λn converging to
Λ such that for every n there exists an immersed J-holomorphic disc fn of class C∞(ID)
centered at f(0), tangent to f at f(0) and satisfying the boundary condition fn(bID) ⊂ Λn.
Proof : Let Λ be an admissible torus and let X be the vector field given by Definition
5.5. In general it is impossible to extend X as a non-vanishing vector field transversal to
f(ID) at every point. However, for any integer (not necessarily positive) n we can consider
the discs hnζ : τ 7→ hζ(ζ
nτ), where ζ ∈ bID. Their tangent vectors at the points f(ζ) are equal
to Xnζ := ζ
nXζ, where by multiplying a vector by a complex number ζ
n we mean applying
the operator (Re ζ + (Im ζ)J)n. We need the following
Lemma 5.7 After a suitable choice of n the vector field Xnζ can be extended on the disc as
a nonvanishing field transversal to f at every point.
Proof : First we look for a global parametrization of a neighborhood of f(ID). Fix an
arbitrary vector field Y transversal to f(ID) at every point. By Nijenhuis - Woolf theorem
we obtain a family of J-holomorphic discs gz : w 7→ gz(w), z ∈ ID so that gz(0) = f(z) and
Xf(ζ) is tangent to gz. Then the map G : (z, w) 7→ gz(w) is a local diffeomorphism from
a neighborhood of ID × ID onto a neighborhood of f(ID) and G(z, 0) = f(z) so we can use
the coordinates (z, w). We pull back the vector field X by G−1 and consider the vector field
(G−1)∗(X) : ζ 7→ (G
−1)∗(Xζ). Let m be the winding number of the w-component of the
vector field (G−1)∗(X) when ζ runs along the circle bID. We set n = −m. Then the field
(G−1)∗(X
n) extends on the disc (ζ, 0) as a smooth vector field Z transversal to this disc at
every point. Then the map G∗(Z) associates to every point of ID a vector transversal to
f(ID) and so defines the desired extension X̃n of the vector field Xn. This proves the lemma.
Now by the Nijenhuis - Woolf theorem there exists a map h̃ζ : ID −→ M which is J-
holomorphic on ID such that h̃ζ = hζ for every ζ in a neighborhood of bID and the vector X̃
is tangent to hζ at the origin. Thus we can extend H to a function defined on ID × ID such
that the map H(ζ, •) is J-holomorphic for any ζ ∈ ID. This map H is a local diffeomorphism
and so determines an admissible parametrization of a neighborhood of f(ID) such that the
generated torus coincides with Λ. Theorem 5.6 now follows by Theorem 5.1.
6 Pushing discs through non-critical levels
In this section we explain how to push a given disc through non-critical level sets of a strictly
plurisubharmonic function.
Proposition 6.1 Suppose that ρ does not have critical values in the closed interval [c1, c2].
Let f : ID −→ Ωc1 be an immersed J-holomorphic disc such that f(bID) ⊂ bΩc1 . Then there
exists an immersed J-holomorphic disc f̃ : ID −→ Ωc2 such that f̃(0) = f(0), df̃(0) = λdf(0)
for some λ > 0 and f̃(bID) ⊂ bΩc2.
For the proof we need some preparations. Let ρ be a strictly plurisubharmonic function
on an almost complex manifold (M,J). For real c consider the domain Ωc = {ρ < c}.
Suppose that its boundary has no critical points. Let f : ID −→ Ωc be a J-holomorphic
disc of class C∞(ID) and such that f(bID) ⊂ bΩc. For every point p ∈ f(bID) consider a
J-holomorphic disc hp : 2ID −→ M touching bΩc from outside such that ρ ◦ hp|2ID\{0} > c.
We call the discs hp the Levi discs. The map hp can be chosen smoothly depending on
p ∈ f(bID).
An explicit construction of the Levi discs is given in [12]. In the almost complex case the
proof is similar; the only thing which has to be justified is the existence of discs hp touching
a strictly pseudoconvex level set from outside. This was recently proved by Barraud and
Mazzilli [1] and Ivashkovich and Rosay [16]. In [6] the result is obtained in any dimension.
For reader’s convenience we include a simple proof (see [6]).
Lemma 6.2 For a point p ∈ bΩc there exists a J-holomorphic disc hp such that hp(0) = p
and hp(ID\{0}) is contained in M\Ωc.
Proof : We fix local coordinates z = (z1, z2) near p such that p = 0 and J(0) = Jst.
Denote by ej , j = 1, 2 the vectors of the standard basis of IC
2. By an additional change of
coordinates we may achieve that the map h : ζ 7→ ζe1 is J-holomorphic on ID. We can
assume that the Levi form LJr (0, e1) = 1 so that
r(z) = 2Re z2 + 2Re
ajkzjzk +
αjkzjzk + o(|z|
α11 = ∆(r ◦ h)(0) = 1.
Now for every δ > 0 consider the non-isotropic dilation Λδ : (z1, z2) 7→ (δ
−1/2z1, δ
−1z2). The
J-holomorphicity of the map h implies that the direct images Jδ := (Λδ)∗(J) converge to Jst
as δ −→ 0 in the Ck norm for every positive integer k on any compact subset of IC2. Similarly,
the functions rδ := δ
−1r ◦ Λ−1 converge to the function r0 := 2Re z2 + |z1|
2 + 2Re βz21 (for
some β ∈ IC).
Consider a Jst-holomorphic disc ĥ : ζ 7→ ζe1 − βζ
2e2. According to the Nijenhuis-
Woolf theorem for every δ ≥ 0 small enough there exists a Jδ-holomorphic discs h
δ such
that the family (hδ)δ≥0 depends smoothly on the parameter δ and for every δ ≥ 0 we have
hδ(ζ) = ζe1 + o(|ζ |) and h
0 = ĥ. Since (r0 ◦ h
0)(ζ) = |ζ |2, we obtain that for δ > 0 small
enough that (rδ ◦ h
δ)(ζ) = Aδ(ζ) + o(|ζ |
2), where Aδ is a positive definite quadratic form on
IR2. Since the structures Jδ and J are biholomorphic, then the lemma follows.
Thus we obtain a smooth map
H : bID× ID −→M, H : (ζ, τ) 7→ hf(ζ)(τ) =: hζ(τ).
For simplicity we assume here that H is a local diffeomorphism although the Levi discs hζ
can intersect even for close values of ζ . We prove in a forthcoming paper that the pullback
H∗(J) of J to the bidisc can be defined even if H is not a local diffeomorphism. Thus
Λ := H(bID × bID) is an admissible torus and ρ|Λ ≥ c + ε for some ε > 0. We stress that ε
depends only on ρ (more precisely on a constant separating the norm of the gradient of ρ
from zero) and the C2-norm of J .
Now Theorem 5.6 implies that there exists a disc f̃ with the same direction as f at the
center and with the boundary attached to a torus arbitrarily close to Λ. Now we cut off the
discs hζ by the level set {ρ = c+ε/2} and obtain a disc with boundary attached to this level
set. Indeed, we have the following
Lemma 6.3 Suppose that ρ ◦ f |bID ≥ c0 and c0 is a non-critical value of ρ. Then there
exists a J-holomorphic disc f̃ centered at f(0) and tangent to f at the center with boundary
attached to the level set {ρ = c0}.
Proof : By the Hopf lemma the disc f intersects the level set {ρ = c0} transversally at
every point. Therefore the open set Ω = {ζ ∈ ID : ρ ◦ f(ζ) < c0} has a smooth bound-
ary. The set Ω may be disconnected, but the connected component of 0 ∈ Ω is simply
connected by the maximum principle applied to the function ρ ◦ f . Now the lemma follows
via reparametrization by the Riemann mapping theorem.
Then we again consider the Levi discs for this level set etc. By iterating this argument
a finite number of times we obtain Proposition 6.1.
7 Pushing discs through a critical level
In order to push the boundary of the disc f through critical level sets of ρ, we use a method
of [11, 7], which consists of temporarily switching to another plurisubharmonic function at
each critical level set. We need a version of the Morse lemma for almost complex manifolds.
Proposition 7.1 Let (M,J) be an almost complex manifold of complex dimension 2. Let
ρ be a strictly plurisubharmonic Morse function on M . Then there exists another strictly
plurisubharmonic Morse function ρ̃ close to ρ with the same critical points, such that at each
critical point of Morse index k in local coordinates given by Lemma 2.2 one has
ρ̃(z) = ρ̃(0) + |z1|
2 + |z2|
2 − a1Re z
1 − a2Re z
2 (17)
where
(i) a1 = a2 = 0 if k = 0,
(ii) a1 = 2 and a2 = 0 if k = 1,
(iii) a1 = a2 = 2 if k = 2.
Remark. This is a weak version of the Morse lemma because we change the given
function ρ instead of reducing it to a normal form.
The following result must be well known. For convenience we include a proof.
Lemma 7.2 Let B be a complex symmetric n×n matrix. Then there exists a unitary matrix
U such that U tBU is diagonal with nonnegative elements.
Proof : Using coordinate-free language, given a hermitian positive definite form H and a
complex symmetric bilinear form B on a vector space V , dimIC V = n, we need u1, ..., un ∈ V
such that
H(ui, uj) = δij , B(ui, uj) = ciδij, ci ≥ 0.
If the above holds with just ci ∈ IC, then by rotation ui 7→ σiui, |σi| = 1, we obtain ci ≥ 0.
It suffices to find u1 ∈ V , H(u1, u1) = 1, such that for every x ∈ V ,
H(x, u1) = 0 implies B(x, u1) = 0. (18)
Then the rest of ui in the H-orthogonal complement of u1 are found by induction. Given
u ∈ V , by duality, there is a unique vector L(u) ∈ V such that for every x ∈ V ,
H(x, L(u)) = B(x, u). (19)
Then L : V → V is a IR-linear (IC-antilinear) transformation. Since B is symmetric, then by
(19), L is real symmetric (self-adjoint) with respect to the form ReH . Then the eigenvalues
of L are real and the eigenvectors are in V (generally they are in V ⊗IR IC). Let u1 ∈ V
be an eigenvector of L, that is L(u1) = λu1, for some λ ∈ IR. We normalize u1 so that
H(u1, u1) = 1. Then for u = u1, (19) implies (18), and the lemma follows.
Proof of Proposition 7.1 : Let p be a critical point of ρ. Introduce a coordinate system
with the origin at p given by Lemma 2.2. In these coordinates the function ρ is strictly
plurisubharmonic at the origin with respect to Jst. Then
ρ(z) = ρ(0) +
aijzizj + Re
bijzizj +O(|z|
where aij = aji and bij = bji. By a linear transformation we can reduce to the form aij = δij .
If we now make a unitary transformation z 7→ Uz preserving |z1|
2 + |z2|
2, then the matrix
B = (bij) changes to U
tBU . By Lemma 7.2 the expression of ρ reduces to
ρ(z) = ρ(0) + |z1|
2 + |z2|
2 − Re (a1z
1 + a2z
2) +O(|z|
where aj ≥ 0, j = 1, 2. The remainder ϕ = O(|z|
3) can be removed by changing ρ to
ρ̃ = ρ−ϕλ, where λ(z) = λ0(z/ε) is a smooth cut-off function with λ0 ≡ 1 in a neighborhood
of the origin and λ0(z) = 0 for |z| ≥ 1, ε > 0 small enough.
Since ϕ(z) = O(|z|3), then |d(ϕλ)| ≤ C|z|2, ‖ ϕλ ‖C2(IC2)≤ Cε where C > 0 is independent
of ε. Since |dρ| ≥ C|z| in a neighborhood of 0 for some C > 0, then for small ε > 0 the
function ρ̃ has only one critical point at the origin, is strictly plurisubharmonic and matches
with ρ for |z| > ε.
The coefficients aj can be reduced to the standard values 0 and 2 depending on the index
k of the critical point. We need a cut-off function that falls down from 1 to 0 sufficiently
slowly.
Lemma 7.3 Given δ > 0 there exists a smooth non-increasing function φ with a compact
support on IR+ such that
(i) φ = 1 near the origin.
(ii) |tφ′(t)| ≤ δ.
(iii) |t2φ′′(t)| ≤ δ
The lemma follows because
Let bj = 0 (resp. 2) if 0 ≤ aj < 1 (resp. aj > 1). Let λ(z) = φ(|z|/ε), where φ is provided
by Lemma 7.3 for sufficiently small δ. Then the function
ρ̃(z) = ρ(z) + λ[(a1 − b1)Re z
1 + (a2 − b2)Re z
for sufficiently small ε has all the desired properties. Proposition 7.1 is proved.
Thus in what follows we assume that ρ has the properties given by Proposition 7.1. Let p
be a critical point of ρ and ρ(p) = 0. Without loss of generality assume that the index k of p
is equal to 1 or 2 since the disc obtained by Proposition 6.1 cannot approach a minimum of
ρ. Choose a small neighborhood U of p. By (17) ρ is strictly plurisubharmonic with respect
to Jst.
We apply the construction of Lemma 6.7 of [11]. Consider c0 > 0 small enough such that
0 is the only critical value of ρ in the interval [−c0, 3c0]. We can assume that c0 is small
enough so that the set K(c0) := {z : ρ(z) ≤ 3c0, |x
′|2 ≤ c0} is compactly contained in a
neighborhood of the origin corresponding to U . Here we use the notation x′ = x1, x
′′ = x2
and |x′|2 = x21 (resp. x
′ = (x1, x2) and |x
′|2 = x21 + x
2 ) if k = 1 (resp. k = 2). We will use
similar notations for the coordinates x, y and the coordinates u, v introduced below. Let
E = {y′ = 0, z′′ = 0, |x′|2 ≤ c0}. (20)
Then E is a totally real submanifold with boundary and dimE = k. Consider the isotropic
dilations of coordinates
dc0 : z 7→ w = u+ iv = c
Set Jc0 = (dc0)∗(J). The structures Jc0 converge to Jst in any C
m norm on compact subsets
of IC2 as c0 −→ 0. Consider the function ρ̂(w) := c
0 ρ(c
0 w). This function has no critical
values in [−1, 3] and its expression in the coordinates w = u+iv is the same as the expression
(17) of ρ that is
ρ̂(w) = 3v21 + v
2 − u
1 + u
if k = 1 and
ρ̂(w) = 3v21 + 3v
2 − u
1 − u
if k = 2. In particular the set K = dc0(K(c0)) is given by {w : ρ̂(w) ≤ 3, |u
′|2 ≤ 1} and is a
fixed compact independent of c0.
It is important that the origin is a critical point of the function ρ and the local coordinates
and the function ρ are given by Proposition 7.1. This allows to use the isotropic dilations in
contrast with Lemma 6.2.
Since the function ρ̂ is strictly plurisubharmonic with respect to Jst, we can apply the
construction of [11] (Lemma 6.7 and section 6.4). We replace the function ρ̂ by a new
function ϕ defined by
ϕ(w) = 3v21 + v
2 − h(u
1) + u
if k = 1 and
ϕ(w) = 3v21 + 3v
2 − h(u
1 + u
if k = 2, where h ≥ 0 is a suitable function. The construction of h depends on the parameter
c0 only. In our “delated” coordinates w we apply this construction taking c0 = 1. Namely,
according to [11] there exist constants 0 < τ0 < τ1 < 1 depending on the eigenvalues of ρ̂
and a function ϕ strictly plurisubharmonic on IC2 with respect to Jst satisfying the following
properties:
(i) ρ̂ ≤ ϕ ≤ ρ̂+ τ1,
(ii) ρ̂+ τ0 ≤ ϕ on the set {|u
′|2 ≥ τ0}
(iii) ϕ = ρ̂+ τ1 on {|u
′|2 ≥ 1}
Since ρ̂ is strictly plurisubharmonic with respect to the structure Jc0 , the function ϕ also is
strictly Jc0-plurisubharmonic on {|u
′|2 ≥ 1} in view of (iii). On the other hand the structures
Jc0 converge to Jst in any C
m norm on compact subsets of IC2 as c0 −→ 0. Therefore, since
ϕ is strictly Jst-plurisubharmonic, it also is strictly Jc0-plurisubharmonic on K if c0 is small
enough. Thus, ϕ is strictly Jc0-plurisubharmonic on {ρ̂ ≤ 3}.
Now consider the function ρ̃(z) = c0ϕ(c
0 z) and set t0 = τ0c0.
The function ρ̃ satisfies the following properties:
(i) ρ̃ is strictly plurisubharmonic (with respect to Jst) in a neighborhood V ⊂ U of 0 and
ρ̃ = ρ+ t1 on the complement of V . Here t1 > 0 is a constant.
(ii) ρ̃ has no critical values on (0, 3c0)
(iii) There exists t0 ∈ (0, c0) such that
{ρ ≤ −c0} ∪ E ⊂ {ρ̃ ≤ 0} ⊂ {ρ ≤ −t0} ∪ E, (21)
where E is defined above by (20).
(iv) We have
{ρ ≤ c0} ⊂ {ρ̃ ≤ 2c0} ⊂ {ρ < 3c0} (22)
By Proposition 6.1 we construct an immersed J-holomorphic disc f such that −t0 <
ρ ◦ f |bID < 0. The boundary of f is contained in a torus Λ formed by discs complex tangent
to a level set of ρ. We will perturb the disc f slightly in order to avoid the intersection of
its boundary with E.
Proposition 7.4 Let f : ID −→ M be an immersed J-holomorphic disc in (M,J), where
dimICM = 2. Let E be a smooth submanifold in M . Then for every m ≥ 2 there exists a
J-holomorphic disc f̃ arbitrarily close to f in Cm(ID) such that f̃(0) = f(0), df̃(0) = df(0),
and f̃ |bID is transverse to E. In particular, if dimIR E ≤ 2, then f̃(bID) ∩ E = ∅.
Proof : By the implicit function theorem, the restriction f |bD admits infinitesimal pertur-
bations in all directions. Then the proposition follows by the proof of Thom’s transversality
theorem.
We now assume f(bID) ∩ E = ∅. In view of the inclusion (21) we conclude that ρ̃ > 0
on f(bID). By Lemma 6.3 we cut off the disc f by a level set {ρ̃ = c} for some c > 0 to
assume that now f(bID) is contained in this level set. The function ρ̃ has no critical values
in (0, 3c0). By Proposition 6.1 applied to the disc f and the function ρ̃ there exists a new
disc f̃ with the boundary contained in {ρ̃ > 2c0}. In view of (22) we have the inclusion
{ρ̃ > 2c0} ⊂ {ρ > c0}. Now the boundary of f̃ is outside the critical level {ρ = 0} as desired,
and we switch back to the original function ρ.
8 Proof of Theorem 1.1
Since the function ρ is strictly plurisubharmonic, then after a generic perturbation of ρ which
does not change the given level set, we can assume that ρ is a Morse function. Let p be the
given point in D. If p is not a point of minimum of ρ, we proceed as follows. Consider a
small J-holomorphic disc f centered at p with the given direction v. Consider a non-critical
level set ρ = c such that ρ(p) < c. Consider a foliation of a neighborhood of f by a complex
one-parameter family of J-holomorphic discs hq, q ∈ f(ID) such that the boundaries of these
discs are outside the sublevel set ρ < c. When q runs over the circle f(bID) these boundaries
form a torus. Applying Proposition 5.1 we obtain a new disc f̃ centered at p and still in the
same direction at p but with ρ ◦ f̃ |bID > 0.
If p is a point of minimum for ρ, we drop this first step and directly have this situation
with f̃ = f . Now the desired results follow by Proposition 6.1 combined with the above
argument allowing to push boundaries of discs through critical levels.
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