On a Linearity Problem for Proper Holomorphic Maps between Balls in Complex Spaces of Different Dimensions

In an important development of several complex variables, Poincaré [26] discovered that any biholomorphic map between two open pieces of the unit sphere in C2 is the restriction of a certain automorphism of B2, the unit two-ball in C2. This phenomenon fails obviously in one complex variable and reveals a strong rigidity property of holomorphic mappings in several variables. Later, Tanaka, etc (see [8], [28]) extended this result to any dimensional case. Alexander, in his famous papers [1], [2], further proved that any proper holomorphic selfmapping of the ball in Cn (n > 1) is an automorphism, thus finishing off a line of research towards the understanding of proper holomorphic mappings between balls in the same complex space. In 1978, using the Cartan-Chern-Moser [8] theory, Webster [31] took up again the problem of considering a proper holomorphic mapping f from the unit n-ball Bn = {z ∈ Cn : |z| < 1} into the unit (n + 1)−ball Bn+1 ⊂ Cn+1 and showed that f is a totally geodesic embedding when f is C3-smooth up to the boundary and when n > 2. Here, we recall that a proper holomorphic map from Bn into BN is called a totally geodesic embedding (or a linear embedding) if there exist automorphisms σ ∈ Aut(Bn) and τ ∈ Aut(BN ) such that τ ◦ f ◦ σ = (id, 0). In a subsequent paper,