Three circle theorem on almost Hermitian manifolds and applications

Abstract

In this paper, we extend Gang Liu's three circle theorem for K\"ahler manifolds to almost Hermitian manifolds. As applications of the three circle theorem, we obtain sharp dimension estimates for spaces of holomorphic functions of polynomial growth and rigidity for the estimates, Liouville theorems for pluri-subharmonic functions of sub-logarithmic growth and Liouville theorems for holomorphic functions of Cheng-type, and fundamental theorem of algebra on almost Hermitian manifolds with slightly negative holomorphic sectional curvature. The results are generalization of Liu's results and some of them are new even for the K\"ahler case. We also discuss the converse of the three circle theorem on Hermitian manifolds which turns out to be rather different with the K\"ahler case. In order to obtain the three circle theorem on almost Hermitian manifolds, we also establish a general maximum principle in the spirit of Calabi's trick so that a general three circle theorem is a straight forward consequence of the general maximum principle. The general maximum principle and three circle theorem established may be of independent interests.