Holomorphic functions on a class of complete Hermitian manifolds

Abstract

In this paper, we prove a gradient estimate for holomorphic functions on complete Hermitian manifolds with nonnegative second Ricci curvature. As an application, we establish several Liouville-type theorems for holomorphic functions and mappings between complete Hermitian manifolds under suitable curvature assumptions. In particular, on a complete Hermitian manifold with nonnegative second Ricci curvature and Chern torsion of linear growth, any bounded holomorphic function must be constant. Moreover, we construct examples to show that both the gradient estimate and this Liouville theorem are sharp in a suitable sense. In the second part of the paper, we study holomorphic functions on simply connected complete Hermitian manifolds with vanishing Chern curvature and torsion norm of sublinear growth. Combining the gradient estimate with methods from sub-Riemannian geometry, we obtain a complete characterization of holomorphic functions of polynomial growth on such manifolds.

arXiv commentIn this version, we change the title and improve the exposition to highlight the main theorem (Theorem 1.1). We also add two examples (Examples 1.3 and 1.4) and update some references