We consider the set of all almost Kähler structures (g,J) on a 2n-dimensional compact orientable manifold M and study a critical point of the functional ${ℱ}_{\lambda ,\mu }(J,g)={\int }_M(\lambda \tau +\mu \tau *)d{M}_g$ with respect to the scalar curvature τ and the *-scalar curvature τ*. We show that an almost Kähler structure (J,g) is a critical point of ${ℱ}_{-1,1}$ if and only if (J,g) is a Kähler structure on M.

Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by $USC{C}_B\left(X\right)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $\phi \in USC{C}_B\left(X\right)$ with its graph which is a closed subset of X × ℝ. The space $USC{C}_B\left(X\right)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $USC{C}_B\left(X\right)$ is homeomorphic to a...