Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds

Abstract

Let D be a J -pseudoconvex region in a smooth almost complex manifold (M, J ) of real dimension four. We construct a local peak J -plurisubharmonic function at every point p ∈ bD of finite D'Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D'Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.

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

“…It was proved in [2] that the (D'Angelo) type and the regular type coincide in four dimensional almost complex manifolds.…”

“…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).…”

“…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.…”

“…In the present paper we study this question for smooth J-pseudoconvex domains of finite type in 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 as…”