On univalence of equivariant Riemann domains over the complexification of a non-compact, Riemannian symmetric space

2006/12/06 by Laura Geatti, Geatti, Laura, Andrea Iannuzzi +1
Mathematics · #32D26 #32M05 #32Q28 #53C35 #Advanced Algebra and Geometry #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math.CV #msc:32D26 #msc:32M05 #msc:32Q28 #msc:53C35

paper · pdf · doi:10.48550/arxiv.math/0612169

46 pages, no figures. v2: minor correction, references updated

openalex publication_date 2006/12/06 · arxiv created 2006/12/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G/K be a non-compact, rank-one, Riemannian symmetric space and let G^C be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over G^C / K^C is necessarily univalent, provided that G is not a covering of SL(2, R). As a consequence of the above statement one obtains a univalence result for holomorphically separable, G x K -equivariant Riemann domains over G^C. Here G x K acts on G^C by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient G^C / K^C, including a complete classification of all its Stein G-invariant subdomains.

Citations

• Cycle Spaces of Flag Domains: A Complex Geometric Viewpoint

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