A note on the geometry of pseudoconvex domains of finite type in almost complex manifolds

Let $D=\{\rho<0\}$ be a smooth domain of finite type in an almost complex manifold (M,J) of real dimension four. We assume that the defining function $\rho$ is J-plurisubharmonic on a neighborhood of $\overline{D}$. We study the asymptotic behavior of pseudoholomorphic discs contained in the domain D.


INTRODUCTION
A well known problem is to determine which smooth domains into an almost complex manifold (M, J) are locally complete hyperbolic in the sense of Kobayashi. It seems natural to make some curvature assumptions on such domains as any J-pseudoconcavity boundary point is at finite Kobayashi distance to the interior of the domain [7]. I.Graham [6] gave asymptotic estimates of the Kobayashi pseudometric for strictly pseudoconvex bounded domains into (C n , J st ) and proved the complete hyperbolicity of those domains. In case (M, J) is any almost complex manifold, similar results were provided by S.Ivashkovich-J.-P.Rosay [7] and H.Gaussier-A.Sukhov [5].
The situation is far from being so clear when the domains are weakly pseudoconvex as the geometry of their boundary is much more complicated; and the question of whether a smooth weakly pseudoconvex domain is locally complete hyperbolic or not is still open, even in (C 2 , J st ). However, D.Catlin [3] obtained local estimates similar to those obtained in [6] on smooth pseudoconvex domains of finite type in (C 2 , J st ), implying their local complete hyperbolicity (see also [1]).
In the present paper we study this question for smooth J-pseudoconvex domains of finite type into a four dimensional almost complex manifold. In [2], we described locally finite type domains D = {ρ < 0} where ρ is a smooth defining function for D and J-plurisubharmonic (see Proposition 2.6 in [2]). More precisely, if D = {ρ < 0} ⊂ C 2 and if the origin 0 ∈ ∂D is of finite type 2m, we proved that there is a change of coordinates in a neighborhood of the origin such that the structure J and the function ρ can be locally written: and, where H 2m is a homogeneous polynomial of degree 2m, subharmonic which is not harmonic and In this paper, we suppose that the harmonic term H(z 1 , z 2 ) in the above expression (0.2) is identically zero and we prove: Theorem 0.1. Let J be a smooth almost complex structure defined on R 4 . Let D = {ρ < 0} be a domain of finite type in (R 4 , J), where ρ is a smooth defining function of D, J-plurisubharmonic in a neighborhood of D. We suppose that J and ρ satisfy respectively (0.1) and (0.2). Moreover we assume that H(z 1 , z 2 ) in (0.2) is identically zero. Then there exists a neighborhood U of the origin for which 0 is at infinite distance from points in D ∩ U .
The proof of this theorem is inspired by [7] and is based on the construction of good J-plurisubharmonic functions whose use is significant in almost complex manifolds.

PRELIMINARIES
We denote by ∆ the unit disc of C and by ∆ r the disc of C centered at the origin of radius r > 0.
1.1. Almost complex manifolds and pseudoholomorphic discs. An almost complex structure J on a real smooth manifold M is a (1, 1) tensor field which satisfies J 2 = −Id. We suppose that J is smooth. The pair (M, J) is called an almost complex manifold. We denote by J st the standard integrable structure on C n for every n. A differentiable map f : The following lemma (see [5]) states that locally any almost complex manifold can be seen as the unit ball of C n endowed with a small smooth perturbation of the standard integrable structure J st . Lemma 1.1. Let (M, J) be an almost complex manifold, with J of class C k , k ≥ 0. Then for every point p ∈ M and every λ 0 > 0 there exist a neighborhood U of p and a coordinate diffeomorphism This is simply done by considering a local chart z : U → B centered a p (ie z(p) = 0), composing it with a linear diffeomorphism to insure z * J (0) = J st and dilating coordinates.
So let J be an almost complex structure defined in a neighborhood U of the origin in R 2n , and such that J is sufficiently closed to the standard structure in uniform norm on the closure U of U . The J-holomorphy equation for a pseudoholomorphic disc u : ∆ → U ⊆ R 2n is given by According to [8], for every p ∈ M , there is a neighborhood V of zero in T p M , such that for every v ∈ V , there is a J-holomorphic disc u satisfying u (0) = p and d 0 u (∂/∂x) = v.

Levi geometry.
Let ρ be a C 2 real valued function on a smooth almost complex manifold (M, J) . We denote by d c J ρ the differential form defined by where v is a section of T M . The Levi form of ρ at a point p ∈ M and a vector v ∈ T p M is defined by The next proposition is useful in order to compute the Levi form (see [7]).
If L J ρ(p, v) ≥ 0 for every p ∈ M and every v ∈ T p M , we say that ρ is J-plurisubharmonic. It is a well know fact that ρ is J-plurisubharmonic if and only if for every J-holomorphic disc u : ∆ → M , ρ • u is subharmonic (see [7]).

Pseudoconvex domains of finite type.
In this section, we recall some facts about pseudoconvex domains of finite type in four dimensional almost complex manifolds (See [2] for more detailed facts). Let Assume that ρ is J-plurisubharmonic on a neighborhood of D where the structure J is defined on a fixed neighborhood U of D. Moreover we suppose that the origin is a boundary point of D. Definition 1.3. Let u : (∆, 0) → R 4 , 0, J be a J-holomorphic disc satisfying u (0) = 0. The order of contact δ 0 (∂D, u) with ∂D at the origin is the degree of the first term in the Taylor expansion of ρ • u. We denote by δ (u) the multiplicity of u at the origin.
We now define the (D'Angelo) type and the regular type of the real hypersurface ∂D at the origin.

Definition 1.4.
(1) The (D'Angelo) type of ∂D at the origin is defined by: The regular type of ∂D at origin is defined by: The type condition as defined in part 1 of Definition 1.4 was introduced by J.-P.D'Angelo [4] who proved that this coincides with the regular type in complex manifolds of dimension two. It was proved in [2] that the (D'Angelo) type and the regular type coincide in four dimensional almost complex manifolds.
In the next proposition, we describe locally the almost complex structure J and the defining function ρ (see [2] for a proof). Proposition 1.5. Let D = {ρ < 0} is a smooth domain in R 4 . Assume that ρ is J-plurisubharmonic on a neighborhood of D where the structure J is defined on a fixed neighborhood U of D. We suppose that the origin 0 ∈ partialD is a point of finite type 2m. Then there is a local change of coordinates in a neighborhood of the origin such that, in the new coordinates: where H 2m is a homogeneous polynomial of degree 2m, subharmonic which is not harmonic and A crucial tool for the study of pseudoholomorphic curves into pseudoconvex domains of finite type is the local peak J-plurisubharmonic functions which existence was proved in [2].
Such a function is called a local peak J-plurisubharmonic function at p. (2) Let D ⊂ M be a domain in an almost complex manifold (M, J). A point p ∈ ∂D is said to be at finite distance from q ∈ D if there is a sequence of points q j ∈ D converging to p and whose Kobayashi distances d (D,J) (q j , q) to q stay bounded. Otherwise we say that the distance is infinite.

PROOF OF THEOREM 0.1
In order to prove this theorem, we need the two following lemmas, where the dimension assumption is meaningful as it allows to find a coordinate system where the lines {z 1 = c} and {z 2 = c ′ } are almost complex submanifolds.

Lemma 2.1.
Let Ω be an open subset of (R 4 , J) and let J be an almost complex structure satisfying (1.2). Let K be a compact subset of Ω. There exists δ > 0 such that: for every r ∈ [0, 1) there exists a positive constant C > 0 such that if u = (u 1 , u 2 ) : ∆ → Ω is a J-holomorphic disc with u(∆) ⊂ K, then This lemma is an anisotropic version of a result obtained by S.Ivashkovich, J.-P.Rosay in [7].
Proof. Depending on u(0) ∈ K, one can make a linear change of variables such that in the new coordinates J(u(0)) = J st . Set α i := sup t∈∆ |u i (t) − u i (0)| for i = 1, 2 and consider the scaling map Λ from R 4 into itself defined by Λ(z 1 , z 2 ) := (α −1 If α 1 and α 2 are small enough then Λ * J is close to J st as the structure J has a diagonal form. It follows from Proposition 2.3.6 in [9], that for |z| ≤ r one gets for i = 1, 2, |∇(Λ • u) i (z)| ≤ C for some positive constant C. Hence |∇u i (z)| ≤ Cα i , as desired.
A straightforward computation leads to this very useful lemma:

Lemma 2.2. Assume that J is an almost complex structure on R 4 satisfying (1.2). Then the Levi form of
Proof of Theorem 0.1. Let U be a neighborhood of 0 in R 4 . We assume that, on U , the structure J satisfies (1.2) and that the defining function has the local expression where H 2m is a homogeneous polynomial of degree 2m, subharmonic which is not harmonic.
Consider for a positive number δ > 0, the following anisotropic polydisc: Notice that since the defining function ρ satisfies (2.1), then for a sufficiently small δ < 1, if z ∈ Q(0, δ) then we have dist (z, ∂D) ≤ cδ for some positive constant c > 0.
Let q ′ = (q ′ 1 , q ′ 2 ) ∈ ∂D ∩ U be a boundary point and let ϕ q ′ be a local peak J-plurisubharmonic function at the point q ′ . There is a positive constant C 1 such that Let u : ∆ → D ∩ U be a J-holomorphic disc such that u (0) ∈ Q(0, δ) is sufficiently close to the origin. In order to prove that the origin 0 is at infinite distance from points in D ∩ U , we want to provide the following estimates |∇u 1 (0) | ≤ Cδ 1 2m and |∇u 2 (0) | ≤ Cδ for a positive constant C > 0.
Let q ′ = (q ′ 1 , q ′ 2 ) ∈ ∂D be the unique boundary point such that q ′ = u(0) + (0, δ u ) for some positive δ u . Notice that δ u is asymptotically equivalent to dist (u (0) , ∂D). According to the J-plurisubharmonicity of Ψ q ′ , we have for |ζ| < r where 0 < r < 1: for an appropriate positive constant A. Hence using (2.2) and the J-plurisubharmonicity of the peak function ϕ q ′ we obtain: Since ) and according to (2.2) and to the fact that u(0) − q ′ = δ u is asymptotically equivalent to dist (u (0) , ∂D), we obtain for a positive constant C 3 : ∂D) . Hence, for some other positive constant C 3 we have: for |ζ| < r where 0 < r < 1.
For the same reason, we obtain the following estimate which hold for |ζ| < r < 1: According to Lemma 2.1, inequalities (2.3) and (2.4) imply: and, for positive constants B and C 4 . Notice that (2.6) is the desired tangential estimate.
In order to obtain the normal estimate |∇u 2 (0)|, we will construct a negative harmonic function. To achieve this, we first need to control ℜeu 2 (ζ) and |∆ℜe u 2 (ζ)| by δ.