Rigidity of the Identity

The structure of the group Aut(Cn) of biholomorphisms of Cn is largely unknown if n > 1. In stark contrast Aut(C) is rather small, consisting of the nonconstant affine linear maps. The description of Aut(C) follows from the observation that an injective holomorphic function f : C → C satisfying f(0) = 0 and f ′(0) = 1 must be the identity. These considerations suggest that similar characterizations of the identity might be useful in understanding the structure of Aut(Cn). Using geometric methods we prove that an injective holomorphic map f : Cn → Cn is the identity I if and only if the power series at 0 of f − I has no terms of order ≤ 2n+1 and the function | det Df(z)| |z|2n|f(z)|−2n is subharmonic throughout Cn. §