For any $n\in\mathbb{N}$ and any $\sigma>1$, there exists a constant $\epsilon=\epsilon(n,\sigma)>0$ depending only on $n$ and $\sigma$ such that, if $(M,\omega,J,g,\Omega)$ is a closed Ricci-flat Calabi-Yau n-manifold with $[\omega]\in H^{2}(M,\mathbb{Z})$, and $p\in M$ such that

• i)

  the injectivity radius and the sectional curvature

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

• ii)

  $[\Omega|_{B_{g}(p,\sigma i_{g}(p))}]\neq 0$ in $H^{n}(B_{g}(p,\sigma i_{g}(p)),\mathbb{C})$,

then there is an open subset $W\subset M$ satisfying that $B_{g}(p,\sigma i_{g}(p))\subset W$, and $(W,\omega,\Omega)$ admits a special lagrangian fibration of a phase $\theta\in\mathbb{R}$, i.e. there is a topological space $B$, and a surjection $f:W\longrightarrow B$ such that, for any $b\in B$, $f^{-1}(b)$ is a smooth n-submanifold,

$$\omega|_{f^{-1}(b)}\equiv 0,\ \ {\rm and}\ \ {\rm Im}e^{\sqrt{-1}\theta}\Omega|_{f^{-1}(b)}\equiv 0.$$