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 $(-1,1)\times \mathbb{R}$. 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.

to appear in Monatshefte fur Mathematik