Differential operators associated with holomorphic mappings
type: Item
instance of: scholarly article
title: Differential operators associated with holomorphic mappings
zbMATH Open document ID: 0830.53009
DOI: 10.1007/BF02108302
author: Karen Pinney Mortensen
author: Robert Molzon
published in: Annals of Global Analysis and Geometry
publication date: 1996-01-28T00:00:00Z
review text: Two differential operators, \(\Sigma\) and \(L\), are studied. They act on holomorphic mappings from domains in \(\mathbb{C}^n\) to \(\mathbb{C} P^n\). \(\Sigma\) is of second order and plays a role in several complex variables similar to that of the Schwarzian (which is of third order) in one variable. The newly introduced operator \(L\) is of third order and is analogous to the Schwarzian considered as a curvature. It is shown that for \(g : B^n \to \mathbb{C} P^n\) locally injective holomorphic, \(n > 1\), \(g\) is projective linear iff \(\Sigma \equiv 0\), and if \(n = 1\) then \(\Sigma\equiv 0\) for all such \(g\). Finally, approximation by projective linear mappings is described.
Mathematics Subject Classification ID: 53A20
Mathematics Subject Classification ID: 53B10
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 729758
zbMATH Keywords: Schwarzian
zbMATH Keywords: projective connection
reviewed by: Alfred Aeppli
MaRDI profile type: Publication
Wikidata QID: Q115392748
cites work: The Schwarzian as a curvature
cites work: The Schwarzian derivative in several complex variables
cites work: Q3135235
cites work: On Uniformization of Complex Manifolds: The Role of Connections. (MN-22)
cites work: Notes on projective structures on complex manifolds
cites work: Q5728083
cites work: The Schwarzian derivative and conformal mapping of Riemannian manifolds
cites work: Projective mappings on differentiable manifolds
cites work: Canonical forms of some systems of linear partial differential equations
full work available at URL: https://doi.org/10.1007/bf02108302
OpenAlex ID: W2054782435
description: scientific article; zbMATH DE number 729758