Several Complex Variables

Abstract. 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 non-constant 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 |detDf(z)||z|²ⁿ|f(z)| ^-2n is subharmonic throughout Cⁿ

Related Publications

[25] F. Xavier Rigidity of the Identity (Comm. in Contemporary Mathematics, to appear)

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