The Schwarzian derivative in Kähler manifolds (Q1906582)

Let \(M\), \(N\) be Kähler manifolds of the same dimension and \(f : M \to N\) be a holomorphic immersion. The authors define the Schwarzian derivative \(S_f\) of \(f\). They prove the following two theorems. Theorem 1. Let \(M\), \(N\), \(H\) be Kähler manifolds with the same complex dimension. Let \(f : M \to N\), \(g : N \to H\) be holomorphic immersions. If \(Sf = 0\), \(Sg = 0\) then \(Sg \circ f = 0\). Theorem 2. Let \(M\) be a complete Kähler manifold, \(\Omega \subseteq M\) be a convex domain, whose holomorphic sectional curvature is bounded above by \(K\). If the real part of the Schwarzian derivative of \(f\) on \(\Omega\) is bounded above by constant \(C(\Omega,K)\) then \(f\) is an embedding.