Pseudoconvex Regions of Finite D’angelo Type in Four Dimensional Almost Complex Manifolds

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. CONTENTS Introduction 1 1. Preliminaries 3 1.1. Almost complex manifolds and pseudoholomorphic discs 3 1.2. Levi geometry 3 2. Construction of a local peak plurisubharmonic function 5 2.1. Pseudoconvex regions of finite D’Angelo type 5 2.2. Construction of a local peak plurisubharmonic function 11 3. Estimates of the Kobayashi pseudometric 15 3.1. The Kobayashi pseudometric 15 3.2. Hyperbolicity of pseudoconvex regions of finite D’Angelo type 15 3.3. Uniform estimates of the Kobayashi pseudometric 18 3.4. Hölder extension of diffeomorphisms 18 4. Sharp estimates of the Kobayashi pseudometric 20 4.1. The scaling method 21 4.2. Complete hyperbolicity in D’Angelo type four condition 24 4.3. Regions with noncompact automorphisms group 27 4.4. Nontangential approach in the general setting 27 5. Appendix : Convergence of the structures involved by the scaling method. 30 References 34 INTRODUCTION Analysis on almost complex manifolds recently became a fondamental tool in symplectic geometry with the work of M.Gromov in [15]. The local existence of pseudoholomorphic discs proved by A.NijenhuisW.Woolf in their paper [21], allows to define the Kobayashi pseudometric, which is crucial for local analysis. In the present paper we study the behaviour of the Kobayashi pseudometric of a J-pseudoconvex region of finite D’Angelo type in an almost complex manifold (M,J) of dimension four. Finite D’Angelo 2000 Mathematics Subject Classification. Primary 32Q60, 32T25, 32T40, 32Q45, 32Q65.