Energy minimizer maps on \(\mathcal C\)-manifolds
type: Item
instance of: scholarly article
title: Energy minimizer maps on \(\mathcal C\)-manifolds
zbMATH Open document ID: 1067.53051
author: Katsuei Kenmotsu
author: Cătălin Gheorghe
published in: Differential Geometry and its Applications
publication date: 2004-08-19T00:00:00Z
review text: A smooth map between manifolds is harmonic if and only if it is a critical point of a natural energy integral which generalizes the Dirichlet integral. Harmonic maps include geodesics, minimal surfaces and holomorphic maps between suitable complex manifolds. A natural question in this setting consists in to wonder when (in what geometric background) a harmonic map minimizes this energy functional. In the paper under review, the authors give a partial answer to this question, showing that certain structure-preserving maps between compact \(c\)-manifolds and Kähler manifolds minimizes the energy functional in its homotopy class. Namely: a \(c\)-manifold is a Riemannian manifold \(M\) where a non-vanishing tensor field \(f\) of type \((1,1)\), (the so-called \(f\)-structure) is defined, in such a way that we have the following main properties: \(f^3+f=0\), \(\text{Ker}\,f\) determines a parallelizable and integrable distribution and the 2-form \(\Omega\) on \(M\) defined by the contraction of \(f\) with the metric determined by the \(f\)-structure, is closed. The product space of a Kähler manifold with an abelian Lie group is a c-manifold. In the paper, the authors: 1) construct non-trivial examples of compact c-manifolds (torus bundles over a Kähler manifold), and 2) show that the smooth maps \(\Psi\) between a compact c-manifold \(M\) and a Kähler manifold \(N\), whose differential \(d\Psi\) intertwines the \(f\)-structure of \(M\) and the Kähler structure \(J\) of \(N\) (i.e., \(d\Psi\circ f=J\circ d\Psi\)) are harmonic maps and, moreover, they are global minimizers of the energy functional in their homotopy class.
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 2092778
zbMATH Keywords: harmonic maps
zbMATH Keywords: energy of maps
zbMATH Keywords: \(f\)-structures
zbMATH Keywords: \(c\)-manifolds
zbMATH Keywords: Kählerian manifolds
reviewed by: Vicente Palmer
MaRDI profile type: Publication
full work available at URL: j.difgeo.2004.03.002
OpenAlex ID: W2089804760
Wikidata QID: Q115358066
cites work: Q4001765
cites work: Geometry of manifolds with structural group \(\mathcal U(n)\times\mathcal O(s)\)
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cites work: Q4938325
cites work: Q5589060
cites work: New examples of compact cosymplectic solvmanifolds
cites work: Q3721461
DOI: 10.1016/J.DIFGEO.2004.03.002
description: scientific article; zbMATH DE number 2092778