I take $F$ from $\Omega\subset \mathbb C^n$ to $\mathbb C^n$ to be a holomorphic function such that
$$| \det(J\_F)|\leq 1,$$
where $J\_F$ is the Jacobian matrix of $F$.


My question: Is there any classification of functions of this type?



Holomorphic function on $\mathbb C^n$
This function has constant Jacobian by Liouville. Then it is
map of Jacobian 1 composed with a homothety or its differential is degenerate everywhere. The constant Jacobian biholomorphisms are subject of much research, see for example


*Rosay, Jean-Pierre,
Automorphisms of Cn, a survey of Andersén-Lempert theory and applications. Complex geometric analysis in Pohang (1997), 131–145,
Contemp. Math., 222, Amer. Math. Soc., Providence, RI, 1999.* 


