Assume that the conclusion is not true. Then, for any fixed $\sigma>1$, there is a family of closed Ricci-flat Calabi-Yau $n$-manifolds $\{(M_{k},\omega_{k},J_{k},g_{k},\Omega_{k})\}$ with $[\omega_{k}]\in H^{2}(M_{k},\mathbb{Z})$, and $p_{k}\in M_{k}$ such that

• i)

  the injectivity radius and the sectional curvature

  $$i_{g_{k}}(p_{k})<\frac{1}{k},\ \ \ \sup_{B_{g_{k}}(p_{k},1)}|K_{g_{k}}|\leq 1,$$

• ii)

  $[\Omega_{k}|_{B_{g_{k}}(p_{k},\sigma i_{g_{k}}(p_{k}))}]\neq 0$ in $H^{n}(B_{g_{k}}(p_{k},\sigma i_{g_{k}}(p_{k})),\mathbb{C})$.

• iii)

  for any open subset $W_{k}^{\prime}\supset B_{g_{k}}(p_{k},\sigma i_{g_{k}}(p_{k}))$, $(W_{k}^{\prime},\omega_{k},\Omega_{k})$ wouldn’t admit special lagrangian fibrations.

If we denote $\tilde{\omega}_{k}=i^{-2}_{g_{k}}(p_{k})\omega_{k}$, $\tilde{g}_{k}=i^{-2}_{g_{k}}(p_{k})g_{k}$, and $\tilde{\Omega}_{k}=i^{-n}_{g_{k}}(p_{k})\Omega_{k}$, then

and $[\tilde{\Omega}_{k}|_{B_{\tilde{g}_{k}}(p_{k},\sigma)}]\neq 0$ in $H^{n}(B_{\tilde{g}_{k}}(p_{k},\sigma),\mathbb{C})$. By the Cheeger-Gromov’s convergence theorem (c.f. Theorem 2.1), a subsequence of $(M_{k},\tilde{\omega}_{k},\tilde{g}_{k},J_{k},\tilde{\Omega}_{k},p_{k})$ converges to a complete flat Calabi-Yau $n$-manifold $(X,\omega_{0},g_{0},J_{0},\Omega_{0},p_{0})$ in the $C^{\infty}$-sense, i.e. for any $r>\sigma$, there are embeddings $F_{r,k}:B_{g_{0}}(p_{0},r)\longrightarrow M_{k}$ such that $F_{r,k}(p_{0})=p_{k}$, and $F_{r,k}^{*}\tilde{g}_{k}$ (resp. $F_{r,k}^{*}\tilde{\omega}_{k}$ and $F_{r,k}^{*}\tilde{\Omega}_{k}$) converges to $g_{0}$ (resp. $\omega_{0}$ and $\Omega_{0}$) in the $C^{\infty}$-sense. Furthermore, $i_{g_{0}}(p_{0})=1$. The soul theorem (c.f. [6], [29]) implies that there is a compact flat totally geodesic submanifold $S\subset X$, the soul, such that $(X,g_{0})$ is isometric to the total space of the normal bundle $\nu(S)$ with a metric induced by $g_{0}|_{S}$ and a natural flat connection.

By Proposition (3.4), there is a finite normal covering $\pi:\tilde{X}\longrightarrow X$ with covering group $\Gamma$ such that

• i)

  $(\tilde{X},\pi^{*}g_{0})$ is isometric to $(T^{n}\times\mathbb{R}^{n},h+h_{E})$, where $T^{n}=\mathbb{R}^{n}/\Lambda$, $\Lambda$ is a lattice in $\mathbb{R}^{n}$, $h_{E}$ is the standard Euclidean metric on $\mathbb{R}^{n}$, and $h$ is the standard flat metric on $T^{n}$ induced by $h_{E}$.

• ii)

  The action of $\Gamma$ on $\tilde{X}$ is a product action, i.e. there are $\Gamma$-actions on $T^{n}$ and $\mathbb{R}^{n}$ such that $\gamma\cdot(x,y)=(\gamma\cdot x,\gamma\cdot y)$ for any $x\in T^{n}$ and $y\in\mathbb{R}^{n}$. Furthermore, $T^{n}\times\{0\}$ is $\Gamma$-invariant, and $S=(T^{n}\times\{0\})/\Gamma$.

• iii)

  $$\pi^{*}\omega_{0}|_{T^{n}\times\{y\}}\equiv 0,\ \ {\rm and}\ \ \pi^{*}{\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{0}|_{T^{n}\times\{y\}}\equiv 0,$$

  for any $y\in\mathbb{R}^{n}$, and a constant $\theta_{0}\in\mathbb{R}$.

Note that the $\Gamma$-action on $\mathbb{R}^{n}$ preserves $h_{E}$, and, $B_{h_{E}}(0,\rho)$, which implies that $\tilde{X}_{\rho}=T^{n}\times B_{h_{E}}(0,\rho)$ are invariant, for any $\rho>0$. Lemma (3.3) shows $[F_{r,k}^{*}\tilde{\omega}_{k}|_{S}]=0$ in $H^{2}(S,\mathbb{R})$, for $k\gg 1$, which implies $[\pi^{*}F_{r,k}^{*}\tilde{\omega}_{k}|_{\tilde{X}_{\rho}}]=0$ in $H^{2}(\tilde{X}_{\rho},\mathbb{R})$, for any $\rho>0$. By Remark (3.5), there are parallel 1-forms $dx_{1},\cdots,dx_{n}$ on $(T^{n},h)$, which are pointwise linear independent, and coordinates $y_{1},\cdots,y_{n}$ on $\mathbb{R}^{n}$ such that

Hence Condition 4.1 is satisfied.

Let $r>\rho\gg\sigma$ such that $B_{g_{0}}(p_{0},\sigma)\subset\pi(\tilde{X}_{\rho})\subset\pi(\tilde{X}_{2\rho})\subset B_{g_{0}}(p_{0},r)$. By Proposition (4.6), for $k\gg 1$, there is an open set $W_{k}\supset\tilde{X}_{\rho}$ such that $(W_{k},\pi^{*}F_{r,k}^{*}\omega_{k},\pi^{*}F_{r,k}^{*}\Omega_{k})$ admits a equivariant special lagrangian fibration $f_{k}:W_{k}\longrightarrow B_{k}$ of phase $\theta_{k}$, where $B_{k}\subset\mathbb{R}^{n}$, i.e. there is a $\Gamma$-action on $B_{k}$, $f_{k}$ is a $\Gamma$-equivariant map, and

for any $b\in B_{k}$. Hence $f_{k}$ induces a special lagrangian fibration $\bar{f}_{k}:\pi(W_{k})\longrightarrow B_{k}/\Gamma$, which implies that $(F_{r,k}\circ\pi(W_{k}),\omega_{k},\Omega_{k})$ admits a special lagrangian fibration, and $F_{r,k}\circ\pi(W_{k})\supset B_{g_{k}}(x_{k},\sigma i_{g_{k}}(x_{k}))$. It is a contradiction. We obtain the conclusion. ∎