Energy minimizer maps on \(\mathcal C\)-manifolds
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.
scientific article; zbMATH DE number 2092778