Higher dimensional generalizations of some theorems on normality of meromorphic functions

In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $mathcal F$ in a domain $Dsubset mathbb C,$ and for a positive constant $epsilon$, if for each $fin mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$min{rho(a_f(z),b_f(z)), rho(b_f(z),c_f(z)), rho(c_f(z),a_f(z))}geq epsilon,$$ for all $zin D$, then $mathcal F$ is normal in $D$. Here, $rho$ is the spherical metric in $widehat{mathbb C}$. In this paper, we establish the high-dimension