The Kobayashi pseudometric for the Fock-Bargmann-Hartogs domain and its application

The Fock-Bargmann-Hartogs domain $D_{n,m}$ in ${\mathbb{C}}^{n+m}$ is defined by the inequality $\mathrm{\parallel }w{\mathrm{\parallel }}^2<e^{-\mathrm{\parallel }z{\mathrm{\parallel }}^2},$ where $(z,w)\in $\mathbb{C}$^n\times $\mathbb{C}$^m$,whichisanunboundednon-hyperbolicdomainin$$\mathbb{C}$^{n+m}$. This paper mainly consists of three parts. Firstly, we give the explicit expression of geodesics of $D_{n,1}$ in the sense of Kobayashi pseudometric; Secondly, using the formula of geodesics, we calculate explicitly the Kobayashi pseudometric on $D_{1,1}$; Lastly, we establish the Schwarz lemma at the boundary for holomorphic mappings between the nonequidimensional Fock-Bargmann-Hartogs domains by using the formula for the Kobayashi pseudometric on $D_{1,1}$.