Parabolic Stein Manifolds

TL;DR

This paper explores various notions of parabolicity for Stein manifolds, relates them to the topological properties of function spaces, and provides examples illustrating these concepts, including complements of zero sets of Weierstrass polynomials.

Contribution

It compiles and compares different definitions of parabolicity for Stein manifolds and connects these to the topological structure of their function spaces, with illustrative examples.

Findings

Multiple notions of parabolicity are related and compared.

The topological type of the function space influences parabolicity.

Examples include complements of zero sets of Weierstrass polynomials with specific exhaustion functions.

Abstract

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among them. In section 3 we relate some of these notions to the linear topological type of the Fr\'echet space of analytic functions on the given manifold. In sections 4 and 5 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.