On holomorphic functions on negatively curved manifolds

Abstract

Based on a well known Sh.-T. Yau theorem we obtain that the real part of a holomorphic function on a Kähler manifold with the Ricci curvature bounded from below by −1 is contractive with respect to the distance on the manifold and the hyperbolic distance on (−1, 1) inhered from the domain (−1, 1) × R. Moreover, in the case of bounded holomorphic functions we prove that the modulus is contractive with respect to the distance on the manifold and the hyperbolic distance on the unit disk.