For any $n\in\mathbb{N}$, there exists a constant $\epsilon=\epsilon(n)>0$ depending only on $n$ such that, if $(M,\omega,J,g)$ is a closed Kähler n-manifold with $[\omega]\in H^{2}(M,\mathbb{Z})$, and

$$M_{\epsilon}=\{p\in M|\ i_{g}(p)<\epsilon,\ \sup_{B_{g}(p,1)}|K_{g}|\leq 1\},$$

then there is an open subset $W\subset M$ such that $W\supset M_{\epsilon}$, and $W$ admits an F-structure $\mathcal{F}$ of positive rank, whose orbits $\mathcal{O}_{p}$, $p\in M_{\epsilon}$, are isotropic submanifolds of $(M,\omega)$, i.e.

$$\omega|_{\mathcal{O}_{p}}\equiv 0.$$