On the Differential Geometric Characterization of The Lee Models

This paper pertains to the J-Hermitian geometry of model domains introduced by Lee (Mich. Math. J. 54(1), 179–206, 2006; J. Reine Angew. Math. 623, 123–160, 2008). We first construct a Hermitian invariant metric on the Lee model and show that the invariant metric actually coincides with the Kobayashi-Royden metric, thus demonstrating an uncommon phenomenon that the Kobayashi-Royden metric is J-Hermitian in this case. Then we follow Cartan’s differential-form approach and find differential-geometric invariants, including torsion invariants, of the Lee model equipped with this J-Hermitian Kobayashi-Royden metric, and present a theorem that characterizes the Lee model by those invariants, up to J-holomorphic isometric equivalence. We also present an all dimensional analysis of the asymptotic behavior of the Kobayashi metric near the strongly pseudoconvex boundary points of domains in almost complex manifolds.