2024/01/26 by G. P. Balakumar, Balakumar, G. P., Jiju Mammen +1
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2401.14700
openalex publication_date 2024/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this article is towards systematically characterizing (holomorphic) retracts of domains of holomorphy; to begin with, bounded balanced pseudoconvex domains B ⊂ ℂN. Specifically, we show that every retract of B passing through its center (origin), is the graph of a holomorphic map over a linear subspace of B. To deal with a case when may fail to have sufficiently many extreme points, we consider products of bounded balanced domains of holomorphy with holomorphically extreme boundaries and obtain a complete description of retracts passing through its center. This can be applied to solve a special case of the union problem with a degeneracy, namely: to characterize those Kobayashi corank one complex manifolds which can be expressed as an increasing union of submanifolds which are biholomorphic to a prescribed homogeneous bounded balanced domain. In this course, we prove a generalization of Mazet's Schwarz lemma. Results about non-existence of retracts of each possible dimension is established for the simplest non-convex but pseudoconvex domain: the ℓq-`ball' for all 0