Holomorphic extensions of maps from the boundary of Kähler manifolds
type: Item
instance of: scholarly article
title: Holomorphic extensions of maps from the boundary of Kähler manifolds
zbMATH Open document ID: 0904.32007
author: Song-Ying Li
author: Jingyi Chen
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-12-29T00:00:00Z
review text: The classical theorem on global holomorphic extensions (of Bochner) asserts that if \(f\) is a smooth function on \(\partial \Omega \), \(\Omega \) be a bounded domain in \({\mathbb C}^n\) with smooth connected boundary, which satisfies the tangential Cauchy-Riemann equation \(\overline{\partial}_bf=0\) on \(\partial \Omega\), then \(f\) can be extended from \(\partial \Omega \) to \(\overline{\Omega}\) so that \(f\) is holomorphic in \( \Omega \). If \(M\) is a complex manifold, then even the local extension problem for a hypersurface is not necessarily solvable in general. Let \(D\) be a domain in a Kaehler manifold \(M\) with smooth compact boundary \(\partial \Omega \). Then \(D\) is called hyperconvex if \(D\) has a smooth defining function with nonnegative trace of the Levi-form on \(\partial D\). According to Siu, a Kaehler manifold \(N\) has strongly seminegative curvature if \(R\) denotes the Riemannian curvature tensor, then \(\sum_{\alpha,\beta,\gamma,\delta}R_{\alpha \overline{\beta}\gamma \overline{\delta}}(A^\alpha \overline{B^\beta}-C^\alpha \overline{D^\beta})(\overline{A^\delta}B^\gamma -\overline{C^\delta}D^\gamma)\) is nonnegative for arbitrary complex numbers \(A^\alpha,B^\alpha,C^\alpha \) and \(D^\alpha \) when \(A^\alpha \overline{B^\beta}-C^\alpha \overline{D^\beta}\neq 0\) for at least one pair of indices \((\alpha,\beta)\). Theorem 1.1. Let \(M(\dim_{{\mathbb C}}M=m>1)\) be a Kaehler manifold and \(D\) a hyperconvex domain in \(M\) with compact closure and non-empty smooth boundary \(\partial D\). Suppose there is a plurisubharmonic function \(\eta \in C^2(\overline{D})\) so that the complex Hessian \(H(\eta)\) has at least two positive eigenvalues at some point in \(D\). Suppose \(N\) is a complete Kaehler manifold with strongly seminegative curvature. Let \(f:\overline{D}\rightarrow N\) be a smooth map satisfying the tangential Cauchy-Riemann equation \(\overline{\partial}_bf=0\) on \(\partial D\). Then there exists a unique holomorphic e…
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32V05
zbMATH DE Number: 1140491
zbMATH Keywords: holomorphic maps
zbMATH Keywords: Kaehler manifolds
zbMATH Keywords: the tangential Cauchy-Riemann equation
zbMATH Keywords: holomorphic extension
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225064
OpenAlex ID: W2065013402
cites work: Analytic and meromorphic continuation by means of Green's formula
cites work: Q4000245
cites work: Harmonic maps of manifolds with boundary
cites work: Two theorems on extensions of holomorphic mappings
cites work: On the extension of holomorphic functions from the boundary of a complex manifold
cites work: On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables
cites work: On the holomorphic equivalence of bounded domains in complete Kaehler manifolds of nonpositive curvature
cites work: Extension of holomorphic maps into hermitian manifolds
cites work: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds
cites work: Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems
cites work: An extension theorem for holomorphic mappings
DOI: 10.2748/TMJ/1178225064
description: scientific article; zbMATH DE number 1140491