Some critical almost Hermitian structures (Q1936907)

Let \(M\) be a compact manifold of even dimension and \(\mathcal{AH}(M)\) the Frechet space of all almost Hermitian structures on \(M\). For a real number \(c>0\), let \(\mathcal{AH}_c(M)\) be the subset of such structures with volume \(c\). For \((\lambda , \mu )\in \mathbb R^2\setminus \{(0, 0)\}\) the functional \(\mathcal{F}_{\lambda , \mu }(J, g)=\int _M(\lambda \tau +\mu \tau ^\ast)dv_g\) is very important where \(\tau \) and \(\tau ^\ast\) are the scalar curvature and \(\ast\)-scalar curvature of \((J, g)\) respectively. The present paper is devoted to the study of the critical points of this functional on the subspaces of \(\mathcal{AH}(M)\) and \(\mathcal{AH}_c(M)\) provided by Hermitian, almost Kähler and nearly Kähler structures respectively. Some very interesting examples are discussed.