Notes on critical almost Hermitian structures

摘要

We discuss the critical points of the functional $mathcal {F}_{lambda, mu} (J, g) = int_M (lambda au + mu au^* ) dv_g$ on the spaces of all almost Hermitian structures $mathcal{AH}(M)$ with ${(lambda, mu)} in mathbb{R}^2 - (0,0)$, where $au$ and $au^*$ being the scalar curvature and the $*$-scalar curvature of $(J, g)$, respectively. We shall give several characterizations of K"{a}hler structure for some special classes of almost Hermitian manifolds, in terms of the critical points of the functionals $mathcal {F}_{lambda, mu} (J, g)$ on $mathcal{AH}(M)$. Further, we provide the almost Hermitian analogy of the Hilbert's result.