A New Proof of the Takeuchi Theorem

In this paper, we present a short proof of the following theorem due to Takeuchi. Theorem A. (Takeuchi [Ta], [Su]) Let Ω be a pseudoconvex domain with C 2-smooth boundary in a Kähler manifold M 2n and r = d(x, bΩ). Suppose that the Kähler manifold M 2n has holomorphic bisectional curvature ≥ 1. Then the second fundamental form of bΩ (−t) satisfies: i∂ ¯ ∂(−r)(ζ, ¯ ζ) ≥ rζ 2 for all ζ ∈ T 1,0 x (bΩ (−t)), where bΩ (−t) = {x ∈ Ω|d(x, bΩ) = t} for t ≥ 0. Moreover, we have the curved version of Oka's inequality: i∂ ¯ ∂(− log r)(ζ, ¯ ζ) ≥ 1 6 ζ 2 for any ζ ∈ T 1,0 x (Ω) and x ∈ Ω. Let us recall some preliminary results. For any C 2 smooth function f and a complex vector τ of (1, 0)-type, the Levi form and complex Hessian are related as follows: Lf (τ, ¯ τ) = 4 n j,k=1 ∂ 2 f ∂z j ∂ ¯ z k τ j ¯ τ k = 2 √ −1(∂ ¯ ∂f)(τ, ¯ τ), where τ = n j=1 τ j ∂ ∂z j