2023/05/03 by Sung‐Yeon Kim, Kim, Sung-Yeon
Mathematics · #14M15 #32H35 #32M15 #32V40 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2305.01875
openalex publication_date 2023/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Dp,q and Dp',q' be irreducible bounded symmetric domains of the first kind with rank q and q', respectively and let f:Dp,q→ Dp',q' be a proper holomorphic map that extends C2 up to the boundary. In this paper we show that if q, q'≥ 2 and f maps Shilov boundary of Dp,q to Shilov boundary of Dp',q', then f is of the form f = \imath∘ F, where F=F1× F2\colon Dp,q→ Ω1'× Ω2', Ω1' and Ω2' are bounded symmetric domains, F1 \colon Dp,q→ Ω1' is a proper rational map, F2:Dp,q→ Ω2' is not proper and \imath: Ω1' × Ω'2 \hookrightarrow Dp',q' is a holomorphic totally geodesic isometric embedding of a reducible bounded symmetric domain Ω1' × Ω2' into Dp',q' with respect to canonical Kähler-Einstein metrics. Moreover, if p>q, then f is a rational map. As an application we show that a proper holomorphic map f:Dp,q→ Dp',q' that extends C^∞ up to the boundary is a rational map or a totally geodesic isometric embedding with respect to the Kobayashi metrics, if 3≤ q ≤ q'≤ 2q-1.