In this paper we discuss some results of the theory of holomorphic almost periodic functions on coverings of complex manifolds, recently developed by the authors. The methods of the proofs are mostly sheaf-theoretic which allows us to obtain new results even in the classical setting of H. Bohr’s holomorphic almost periodic functions on tube

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow...

The Goursat representation formula in the complex plane, expressing a real–valued biharmonic function in terms of two holomorphic functions and their anti–holomorphic complex conjugates, is generalized to Euclidean space, expressing a real–valued polyharmonic function of order p in terms of p so–called monogenic functions of Clifford analysis. Mathematics Subject Classification (2010). 30G35.

We apply the methods developed in [Br1] to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin [GHS] on holomorphic L2 functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.

We will develop some of the basic concepts of complex function theory and prove a number of useful results concerning holomorphic functions. We will focus on derivatives, zeros, and sequences of holomorphic functions. This will lead to a brief discussion of the significance of biholomorphic mappings and allow us to prove the Riemann mapping theorem.

LetG be a connected affine algebraic reductive group over C. Let P be a holomorphic principal G bundle on M (i.e. the transition functions of P are holomorphic). Assume that P admits a holomorphic connection D compatible with the holomorphic structure. This means the following : D is a holomorphic 1-form on P with values in the Lie algebra, g, of G, such that D is invariant for the action of G ...

[03.1] For a bounded sequence of complex numbers c n , prove that ∞ n=0 c n z n z n + 1 converges to a holomorphic function on |z| < 1. Each summand is holomorphic on |z| < 1, because of the quotient rule, and that the numerator and denominator are polynomials, hence holomorphic. To prove that the sum n f n of a sequence of holomorphic functions on |z| < 1 is itself holomorphic, it suffices to ...

Let F be a collection of holomorphic functions and let R(PR(F)) denote the reduct of the structure Ran to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in F . We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure R(...

We study divisibilities between bounded holomorphic quasi-inner functions in H∞ and operator-valued bounded holomorphic quasi-inner functions in H∞(Ω, L(K)) where Ω is a bounded finitely connected region. Furthermore, we characterize these divisibilities by using rationally-invariant subspaces.