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

TL;DR

This paper constructs local peak J-plurisubharmonic functions at boundary points of finite D'Angelo type in four-dimensional almost complex manifolds, leading to local Kobayashi hyperbolicity and sharp metric estimates.

Contribution

It introduces a method to construct peak functions at boundary points of finite D'Angelo type in four-dimensional almost complex manifolds, advancing understanding of local hyperbolicity.

Findings

Constructed local peak J-plurisubharmonic functions at boundary points.

Established local estimates of the Kobayashi pseudometric.

Proved local Kobayashi hyperbolicity at boundary points of finite D'Angelo type.

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 boundary point p 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.