Biharmonic holomorphic maps and conformally Kähler geometry
type: Item
instance of: scholarly article
title: Biharmonic holomorphic maps and conformally Kähler geometry
zbMATH Open document ID: 1315.58010
published in: Israel Journal of Mathematics
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.2104
review text: The aim of this article is to provide a method for constructing biharmonic maps from holomorphic maps. Let \(\phi\) be a map between two Riemannian manifolds, then its tension field is given by \(\tau(\phi):=\text{trace}\nabla d\phi\). Biharmonic maps are critical points of the \(L^2\)-norm of the tension field, the corresponding Euler-Lagrange equation is a fourth order partial differential equation. Motivated by the Lichnerowicz theorem for harmonic maps, the authors derive conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map into a \((1,2)\)-symplectic manifold is biharmonic. These conditions turn out to be non-linear and of third order. However, it is shown that the latter simplify on locally conformal Kähler manifolds. In addition, the authors provide construction methods and give examples in arbitrary dimensions.
reviewed by: Volker Branding
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53B35
zbMATH DE Number: 6361308
zbMATH Keywords: biharmonic maps
zbMATH Keywords: local conformal Kähler manifold
zbMATH Keywords: Lee vector field
zbMATH Keywords: holomorphic map
author: M. Benyounes
author: Eric Loubeau
author: Radu Slobodeanu
MaRDI profile type: Publication
OpenAlex ID: W2091617606
arXiv ID: 1204.2104
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DOI: 10.1007/S11856-014-1029-Z
description: scientific article; zbMATH DE number 6361308