Sharp estimates of the Kobayashi metric and Gromov hyperbolicity

Let D = {ρ < 0} be a smooth relatively compact domain in a four-dimensional almost complex manifold (M, J ), where ρ is a J-plurisubharmonic function on a neighborhood of D and strictly J-plurisubharmonic on a neighborhood of ∂D. We give sharp estimates of the Kobayashi metric. Our approach is based on an asymptotic quantitative description of both the domain D and the almost complex structure J near a boundary point. Following Z.M. Balogh and M. Bonk [Z.M. Balogh, M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000) 504–533], these sharp estimates provide the Gromov hyperbolicity of the domain D.