Polynomials on Parabolic Manifolds

TL;DR

This paper studies polynomials on S-parabolic Stein manifolds, exploring their properties, examples, and the relationship between polynomial density and manifold regularity, with implications for the structure of analytic function spaces.

Contribution

It introduces a new classification of S-parabolic manifolds based on polynomial density and regularity, and analyzes the topological properties of their function spaces.

Findings

Constructed an example of an S-parabolic manifold with no nontrivial polynomials.

Established a division of S-parabolic manifolds into regular and non-regular based on polynomial density.

Proved that on regular S-parabolic manifolds, the space of analytic functions has a polynomial basis.

Abstract

A Stein manifold X is called S-parabolic if it possesses a plurisub- harmonic exhaustion function p that is maximal outside a compact subset of X: In analogy with (Cn; ln jzj), one defines the space of polynomials on a S- parabolic manifold (X; p) as the set of all analytic functions with polynomial growth with respect to p. In this work, which is, in a sense continuation of [7], we will primarily study polynomials on S-parabolic Stein manifolds. In Section 2, we review different notions of paraboliticity for Stein manifolds, look at some examples and go over the connections between parabolicity of a Stein manifold X and certain linear topological properties of the Fr\'echet space of global analytic functions on X. In Section 4 we construct an example of a S-parabolic manifold, with no nontrivial polynomials. This example leads us to divide S-parabolic manifolds into two groups as the ones whose class of polynomials is dense in the corresponding space of analytic functions and the ones whose class of polynomials is not so rich. In this way we introduce a new notion of regularity for S-parabolic manifolds. In the final section we investigate linear topolog- ical properties of regular S-parabolic Stein manifolds and show in particular that the space of analytic functions on such manifolds have a basis consisting of polynomials. We also give a criterion for closed submanifolds of a regular S-parabolic to be regular S-parabolic, in terms of existence of tame extension operators for the spaces of analytic functions defined on these submanifolds.