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Let Bn+1 ⊂ Cn+1 be the unit ball in a complex Euclidean space, and let n = ∂Bn+1 = S2n+1. Let f : n ֒→ N be a local CR immersion. If N −n < 2n−1, the asymptotic vectors of the second fundamental form of f at each point form a subspace of the holomorphic tangent space ofn of codimension at most 1. We exploit the successive derivatives of this relation and show that a linearly full local CR immersion f : n ֒→ N, N ≤ 3n − 2, can only occur when N = n, 2n, or 2n + 1. Together with the recent classification of the rational proper holomorphic maps from Bn+1 to B2n+2 by Hamada, this gives a classification of the rational proper holomorphic maps from Bn+1 to B3n−1 for n ≥ 3.

RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1TO B3n-1

RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bⁿ⁺¹TO B^3n-1