Reinhardt domains determined by their endomorphisms

TL;DR

This paper demonstrates that certain Reinhardt domains and Stein manifolds are uniquely characterized by their semigroup of holomorphic endomorphisms, up to biholomorphic or anti-biholomorphic equivalence.

Contribution

It establishes that pseudoconvex Reinhardt domains in two dimensions and specific Stein manifolds are uniquely determined by their endomorphism semigroups.

Findings

Reinhardt domains with isomorphic endomorphism semigroups are equivalent.

Stein manifolds retracting to punctured complex lines are characterized by their endomorphism semigroups.

Abstract

We show that pseudoconvex Reinhardt domains in dimension two with isomorphic semigroups of holomorphic endomorphisms are biholomorphically or anti-biholomorphically equivalent. Moreover, we show that every Stein manifold that retracts to a properly embedded copy of the punctured complex line, is determined (up to biholomorphic or anti-biholomorphic equivalence) by its semigroup of holomorphic endomorphisms.