4. Local special lagrangian fibrations [05DS]

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4. Local special lagrangian fibrations

In this section, we study the deformation of special lagrangian fibrations under the convergence of Calabi-Yau metrics. Let $(Y,\omega,g,J,\Omega)$ be a complete flat Calabi-Yau $n$ -manifold.

Condition 4.1.

Assume that

i)

$Y=T^{n}\times\mathbb{R}^{n}$ , $g=h+h_{E}$ , and the natural projection $f:Y\longrightarrow\mathbb{R}^{n}$ is a special lagrangian fibration of $(Y,\omega,\Omega)$ , where $T^{n}=\mathbb{R}^{n}/\Lambda$ is a torus, $\Lambda$ 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 induced by $h_{E}$ .

ii)

We assume that 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

g=h+h_{E}=\sum(dx_{j}^{2}+dy_{j}^{2}),\ \ \ \omega=\sum dx_{j}\wedge dy_{j},\ \ \ \Omega=\bigwedge_{j=1}^{n}(dx_{j}+\sqrt{-1}dy_{j}).

iii)

There is a family of Calabi-Yau structures $(\omega_{k},g_{k},J_{k},\Omega_{k})$ converging to $(\omega,g,J,\Omega)$ in the $C^{\infty}$ -sense on $Y_{2r}=T^{n}\times B_{h_{E}}(0,2r)$ for a $r\gg 1$ , where $B_{h_{E}}(0,2r)=\{y\in\mathbb{R}^{n}|\|y\|_{h_{E}}<2r\}$ . Moreover, $\omega_{k}\in[\omega]\in H^{2}(Y_{2r},\mathbb{R})$ .
vi)

There is a finite group $\Gamma$ acting on $Y_{2r}$ preserving $\omega_{k},g_{k},\Omega_{k},\omega,g,\Omega$ , and $T^{n}\times\{0\}$ is a invariant set. The $\Gamma$ -action is a product action on $T^{n}\times B_{h_{E}}(0,2r)$ . The natural projection $f:Y\longrightarrow\mathbb{R}^{n}$ is $\Gamma$ -equivariant.

The goal of this section is to construct equivariant special lagrangian fibrations on $(Y_{r},\omega_{k},\Omega_{k})$ for $k\gg 1$ .

Denote $L=T^{n}\times\{0\}$ , which is a special lagrangian submanifold of $(Y,\omega,\Omega)$ , i.e. $\omega|_{L}=0$ and ${\rm Im}\Omega|_{L}=0$ . Note that we can identify $Y$ with the total space of the normal bundle $\nu(L)$ by the exponential map from $\nu(L)$ to $Y$ , $\exp_{L,g}:(x,\sum_{j}y_{j}\frac{\partial}{\partial y_{j}})\mapsto(x,y)$ where $x\in L$ and $y=(y_{1},\cdots,y_{n})$ . There is a canonical bundle isomorphism from $\nu(L)$ to the cotangent bundle $T^{*}L$ given by $v\mapsto\iota(v)\omega$ where $v\in\nu_{x}(L)$ . Thus we can identify $Y$ with the total space of $T^{*}L$ by the map

(1)

(x,y)\mapsto(x,\iota(\sum_{j}y_{j}\frac{\partial}{\partial y_{j}})\omega)=(x,\sum_{j}y_{j}dx_{j}),

where $x\in L$ and $y=(y_{1},\cdots,y_{n})\in\mathbb{R}^{n}$ . We do not distinguish $Y$ with $T^{*}L$ in this section for convenience. For a 1-form $\sigma$ on $L$ , and a $y\in\mathbb{R}^{n}$ , which can be regarded as a 1-form from above,

L(y,\sigma)=\{(x,y+\sigma(x))|x\in L\}

denotes the graph of $y+\sigma$ , i.e. $y=\sum y_{j}dx_{j}$ , $\sigma=\sum\sigma_{j}dx_{j}$ , and

L(y,\sigma)=\{(x,y_{1}+\sigma_{1}(x),\cdots,y_{n}+\sigma_{n}(x))|x\in L\}.

There are two constants $a_{k}>0$ and $\theta_{k}\in\mathbb{R}$ , for any $k$ , such that

\int_{L}\Omega_{k}=a_{k}e^{-\sqrt{-1}\theta_{k}}\int_{L}\Omega_{0}=a_{k}e^{-\sqrt{-1}\theta_{k}}\int_{L}{\rm Re}\Omega_{0},

$\lim\limits_{k\longrightarrow\infty}a_{k}=1$ and $\lim\limits_{k\longrightarrow\infty}\theta_{k}=0$ by the smooth convergence of $\Omega_{k}$ . There are real 1-forms $\alpha_{k}$ and complex value $(n-1)$ -forms $\beta_{k}$ such that

\omega_{k}=\omega_{0}-d\alpha_{k},\ \ \ \Omega_{k}=a_{k}e^{-\sqrt{-1}\theta_{k}}(\Omega_{0}+d\beta_{k})

by $\omega_{k}\in[\omega]\in H^{2}(Y_{2r},\mathbb{R})$ . By the smooth convergence of $\omega_{k}$ and $\Omega_{k}$ ,

(2)

\lim_{k\longrightarrow\infty}\|d\alpha_{k}\|_{C^{2}(Y_{2r},g)}=\lim_{k\longrightarrow\infty}\|d\beta_{k}\|_{C^{2}(Y_{2r},g)}=0.

Define a diffeomorphism $\Pi:L\longrightarrow L(y,\sigma)$ by $x\mapsto(x,y+\sigma(x))$ for a $y\in\mathbb{R}^{n}$ and a 1-form $\sigma$ on $L$ . If

(3)

\mathfrak{F}_{k}(y,\sigma)=(-\Pi^{*}\omega_{k}|_{L(y,\sigma)},*_{h}a_{k}^{-1}\Pi^{*}{\rm Im}e^{\sqrt{-1}\theta_{k}}\Omega_{k}|_{L(y,\sigma)}),

where $*_{h}$ is the Hodge star operator on $(L,h)$ , then $L(y,\sigma)$ is a special lagrangian submanifold of $(Y,\omega_{k},\Omega_{k})$ of phase $\theta_{k}$ if and only if

\mathfrak{F}_{k}(y,\sigma)=0.

A straightforward calculation (c.f. [28]) gives

(4)

\mathfrak{F}_{k}(y,\sigma)=(d\sigma+\Pi^{*}d\alpha_{k}|_{L(y,\sigma)},*_{h}d*_{h}\sigma+*_{h}\Pi^{*}d{\rm Im}\beta_{k}|_{L(y,\sigma)}).

We denote $\Omega^{j}(L)$ the space of $j$ -forms on $L$ , and define two Banach spaces $\mathfrak{B}_{1}=C^{1,\alpha}(d\Omega^{0}(L)\oplus d^{*_{h}}\Omega^{2}(L))$ and $\mathfrak{B}_{2}=C^{0,\alpha}(d\Omega^{1}(L)\oplus d^{*_{h}}\Omega^{1}(L))$ . Then $\mathfrak{F}_{k}$ defines a smooth map $\mathfrak{F}_{k}:\mathcal{U}(r)\longrightarrow\mathfrak{B}_{2}$ for any $k$ , where $\mathcal{U}(r)=\{\|y\|_{h_{E}}+\|\sigma\|_{C^{1,\alpha}(L,h)}<2r|(y,\sigma)\in\mathbb{R}^{n}\times\mathfrak{B}_{1}\}$ .

Lemma 4.2.

For any $y\in B_{h_{E}}(0,2r)$ ,

\|\mathfrak{F}_{k}(y,0)\|_{C^{0,\alpha}(L,h)}\leq C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)},

for a constant $C$ independent of $k$ .

Proof.

Since

\mathfrak{F}_{k}(y,0)=(\Pi^{*}d\alpha_{k}|_{L(y,0)},*_{h}\Pi^{*}d{\rm Im}\beta_{k}|_{L(y,0)}),

we obtain the conclusion by straightforward calculations. ∎

The differentials of $\mathfrak{F}_{k}(y,\sigma)$ are

(5)

D_{\sigma}\mathfrak{F}_{k}(y,\sigma)\dot{\sigma}=(d\dot{\sigma},*_{h}d*_{h}\dot{\sigma})+(D_{\sigma}(\Pi^{*}d\alpha_{k}|_{L(y,\sigma)})\dot{\sigma},*_{h}D_{\sigma}(\Pi^{*}d{\rm Im}\beta_{k}|_{L(y,\sigma)})\dot{\sigma}),

(6)

D_{y}\mathfrak{F}_{k}(y,\sigma)\dot{y}=(D_{y}(\Pi^{*}d\alpha_{k}|_{L(y,\sigma)})\dot{y},*_{h}D_{y}(\Pi^{*}d{\rm Im}\beta_{k}|_{L(y,\sigma)})\dot{y})

{\rm and}\ \ \ D\mathfrak{F}_{k}(y,\sigma)(\dot{y}+\dot{\sigma})=D_{\sigma}\mathfrak{F}_{k}(y,\sigma)\dot{\sigma}+D_{y}\mathfrak{F}_{k}(y,\sigma)\dot{y}.

Under the frame field $dx_{1},\cdots,dx_{n}$ and coordinates $y_{1},\cdots,y_{n}$ ,

d\alpha_{k}=\sum_{ij}(\alpha_{k,ij}dx_{i}\wedge dx_{j}+\alpha_{k,i(n+j)}dx_{i}\wedge dy_{j}+\alpha_{k,(n+i)(n+j)}dy_{i}\wedge dy_{j}).

The differential is

$\displaystyle D(\Pi^{*}d\alpha_{k}\|_{L(y,\sigma)})(\dot{y}+\dot{\sigma})$	$\displaystyle=$	$\displaystyle\sum_{ijl}(\frac{\partial\alpha_{k,ij}}{\partial y_{l}}(\dot{y}_{l}+\dot{\sigma}_{l})dx_{i}\wedge dx_{j}+\alpha_{k,i(n+j)}dx_{i}\wedge d\dot{\sigma}_{j}$
		$\displaystyle+\frac{\partial\alpha_{k,i(n+j)}}{\partial y_{l}}(\dot{y}_{l}+\dot{\sigma}_{l})dx_{i}\wedge d\sigma_{j}+\alpha_{k,(n+i)(n+j)}d\sigma_{i}\wedge d\dot{\sigma}_{j}$
		$\displaystyle\frac{\partial\alpha_{k,(n+i)(n+j)}}{\partial y_{l}}(\dot{y}_{l}+\dot{\sigma}_{l})d\sigma_{i}\wedge d\sigma_{j}).$

We obtain

(7)

\|D_{\sigma}(d\alpha_{k}|_{L})\dot{\sigma}\|_{C^{0,\alpha}(L,h)}\leq C\|d\alpha_{k}\|_{C^{1,\alpha}(Y_{2r},g)}\|\dot{\sigma}\|_{C^{1,\alpha}(L,h)},

\|D_{\sigma}(\Pi^{*}d\alpha_{k}|_{L(y,\sigma)})\dot{\sigma}\|_{C^{0,\alpha}(L,h)}\leq C\|d\alpha_{k}\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,2}\|\sigma\|_{C^{1,\alpha}(L,h)}^{l})\|\dot{\sigma}\|_{C^{1,\alpha}(L,h)},

\|D_{y}(\Pi^{*}d\alpha_{k}|_{L(y,\sigma)})\dot{y}\|_{C^{0,\alpha}(L,h)}\leq C\|d\alpha_{k}\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,2}\|\sigma\|_{C^{1,\alpha}(L,h)}^{l})\|\dot{y}\|_{h_{E}},

for a constant $C$ independent of $k$ . The same argument gives

(8)

\|D_{\sigma}(d{\rm Im}\beta_{k}|_{L})\dot{\sigma}\|_{C^{0,\alpha}(L,h)}\leq C\|d\beta_{k}\|_{C^{1,\alpha}(Y_{2r},g)}\|\dot{\sigma}\|_{C^{1,\alpha}(L,h)},

\|D_{\sigma}(\Pi^{*}d{\rm Im}\beta_{k}|_{L(y,\sigma)})\dot{\sigma}\|_{C^{0,\alpha}(L,h)}\leq C\|d\beta_{k}\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,\cdots,n}\|\sigma\|_{C^{1,\alpha}(L,h)}^{l})\|\dot{\sigma}\|_{C^{1,\alpha}(L,h)},

\|D_{y}(\Pi^{*}d{\rm Im}\beta_{k}|_{L(y,\sigma)})\dot{y}\|_{C^{0,\alpha}(L,h)}\leq C\|d\beta_{k}\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,\cdots,n}\|\sigma\|_{C^{1,\alpha}(L,h)}^{l})\|\dot{y}\|_{h_{E}}.

Lemma 4.3.

The operator $D_{\sigma}\mathfrak{F}_{k}(0,0)$ is invertible for $k\gg 1$ , and

\|D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}\|\leq\overline{C},

for a constant $\overline{C}>0$ independent of $k$ .

Proof.

Note that

D_{\sigma}\mathfrak{F}_{k}(0,0)\dot{\sigma}=(d\dot{\sigma},*_{h}d*_{h}\dot{\sigma})+(D_{\sigma}(d\alpha_{k}|_{L})\dot{\sigma},*_{h}D_{\sigma}(d{\rm Im}\beta_{k}|_{L})\dot{\sigma})=(\mathcal{D}+V_{k})\dot{\sigma},

where $\mathcal{D}=d-*_{h}d*_{h}$ is the restriction of the Hodge Dirac operator $d+d^{*_{h}}$ on the space of 1-forms, and, thus, is an elliptic operator of 1-order. By the standard elliptic estimate (c.f. Proposition 1.5.2 in [23] and [20]), we have

\|\xi\|_{C^{1,\alpha}(L,h)}\leq C_{S}\|\mathcal{D}\xi\|_{C^{0,\alpha}(L,h)},

for any $\xi\in\mathfrak{B}_{1}$ , and a constant $C_{S}$ independent of $k$ . Hence $\mathcal{D}$ is injective. From the definition of $\mathfrak{B}_{2}$ , $\mathcal{D}$ is also surjective, which implies that $\mathcal{D}$ is invertible from $\mathfrak{B}_{1}$ to $\mathfrak{B}_{2}$ . Moreover,

\|\mathcal{D}^{-1}\|\leq C_{S}.

By (7) and (8),

\|V_{k}\|\leq C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)}<\frac{1}{2C_{S}},

for $k\gg 1$ , and, thus,

\|\mathcal{D}^{-1}V_{k}\|<\frac{1}{2}.

By the standard operator’s theory (c.f. [36]), $D_{\sigma}\mathfrak{F}_{k}(0,0)=\mathcal{D}+V_{k}$ is invertible, and the inverse operator is defined by

D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}=(\sum_{j=0}^{\infty}(-1)^{j}(\mathcal{D}^{-1}V_{k})^{j})\mathcal{D}^{-1}.

We obtain

\|D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}\|\leq(\sum_{j=0}^{\infty}2^{-j})\|\mathcal{D}^{-1}\|\leq\overline{C},

for a constant $\overline{C}>0$ independent of $k$ . ∎

Lemma 4.4.

For any $\delta_{0}\ll 1$ , there is a constant $k_{0}\gg 1$ such that, if $\|y\|_{h_{E}}\leq\frac{3r}{2}$ and $\|\sigma\|_{C^{1,\alpha}(L,h)}\leq\delta_{0}$ , and $k>k_{0}$ , then

\|D_{\sigma}\mathfrak{F}_{k}(y,\sigma)-D_{\sigma}\mathfrak{F}_{k}(0,0)\|\leq\frac{1}{2\overline{C}}.

Furthermore, $D_{\sigma}\mathfrak{F}_{k}(y,\sigma)$ is also invertible, and

\|D_{\sigma}\mathfrak{F}_{k}(y,\sigma)^{-1}\|\leq 2\overline{C}.

Proof.

By (5),

	$\displaystyle(D_{\sigma}\mathfrak{F}_{k}(y,\sigma)-D_{\sigma}\mathfrak{F}_{k}(0,0))\dot{\sigma}$	$\displaystyle=$	$\displaystyle((D_{\sigma}(\Pi^{*}d\alpha_{k}\|_{L(y,\sigma)})-D_{\sigma}(d\alpha_{k}\|_{L}))\dot{\sigma},$
			$\displaystyle_{h}(D_{\sigma}(\Pi^{}d{\rm Im}\beta_{k}\|_{L(y,\sigma)})-D_{\sigma}(d{\rm Im}\beta_{k}\|_{L}))\dot{\sigma}).$

We can take a $k_{0}\gg 1$ such that, for $k>k_{0}$ ,

$\displaystyle\\|D_{\sigma}\mathfrak{F}_{k}(y,\sigma)-D_{\sigma}\mathfrak{F}_{k}(0,0)\\|$	$\displaystyle\leq$	$\displaystyle 2C\\|(d\alpha_{k},d\beta_{k})\\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,\cdots,n}\\|\sigma\\|_{C^{1,\alpha}(L,h)}^{l})$
	$\displaystyle\leq$	$\displaystyle 2C\\|(d\alpha_{k},d\beta_{k})\\|_{C^{1,\alpha}(Y_{2r},g)}n\delta_{0}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{4\overline{C}},$

by (2), (7) and (8). We obtain the first formula in the conclusion.

Note that $D_{\sigma}\mathfrak{F}_{k}(y,\sigma)=D_{\sigma}\mathfrak{F}_{k}(0,0)+(D_{\sigma}\mathfrak{F}_{k}(y,\sigma)-D_{\sigma}\mathfrak{F}_{k}(0,0))$ , $D_{\sigma}\mathfrak{F}_{k}(0,0)$ is invertible, and $\|D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}\|\leq\overline{C}$ . By the same arguments as in the proof of Lemma 4.3, and

\|D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}(D_{\sigma}\mathfrak{F}_{k}(y,\sigma)-D_{\sigma}\mathfrak{F}_{k}(0,0))\|\leq\frac{1}{2},

$D_{\sigma}\mathfrak{F}_{k}(y,\sigma)$ is also invertible, and

\|D_{\sigma}\mathfrak{F}_{k}(y,\sigma)^{-1}\|\leq(\sum_{j=0}^{\infty}2^{-j})\|D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}\|\leq 2\overline{C}.

∎

Lemma 4.5.

For a fixed $\delta<\delta_{0}$ , there is a $k_{1}>k_{0}$ such that, for any $y\in B_{h_{E}}(0,\frac{3r}{2})$ and $k>k_{1}$ , there is a unique $\sigma_{k}(y)\in\mathfrak{B}_{1}$ , such that

\mathfrak{F}_{k}(y,\sigma_{k}(y))=0,\ \ \ \|\sigma_{k}(y)\|_{C^{1,\alpha}(L,h)}\leq\delta,

which implies that $L(y,\sigma_{k}(y))$ is a special lagrangian submanifold of $(Y_{2r},\omega_{k},\Omega_{k})$ . Furthermore,

\|D\sigma_{k}(y)\|\leq 2n\delta\overline{C}C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)},

for a constant $C$ independent of $k$ .

Proof.

Fix a $\delta<\delta_{0}$ , there is a $k_{1}>k_{0}$ such that, for $k>k_{1}$ , and any $y\in B_{h_{E}}(0,2r)$ ,

\|\mathfrak{F}_{k}(y,0)\|_{C^{0,\alpha}(L,h)}\leq\frac{\delta}{4\overline{C}},

by Lemma 4.2. By Theorem 2.3, Lemma 4.3 and 4.4, for any $y\in B_{h_{E}}(0,\frac{3r}{2})$ and $k>k_{1}$ , there is a unique $\sigma_{k}(y)\in\mathfrak{B}_{1}$ such that

(9)

\mathfrak{F}_{k}(y,\sigma_{k}(y))=0,\ \ \ \|\sigma_{k}(y)\|_{C^{1,\alpha}(L,h)}\leq\delta,

which implies that $L(y,\sigma_{k}(y))$ is a special lagrangian submanifold of $(Y_{2r},\omega_{k},\Omega_{k})$ .

By (6) (7) and (8),

	$\displaystyle\\|D_{y}\mathfrak{F}_{k}(y,\sigma_{k})\\|$	$\displaystyle\leq$	$\displaystyle C\\|(d\alpha_{k},d\beta_{k})\\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,\cdots,n}\\|\sigma_{k}\\|_{C^{1,\alpha}(L,h)}^{l})$
		$\displaystyle\leq$	$\displaystyle C\\|(d\alpha_{k},d\beta_{k})\\|_{C^{1,\alpha}(Y_{2r},g)}n\delta,$

for a constant $C$ independent of $k$ . By Theorem 2.3,

D\sigma_{k}(y)\dot{y}=-D_{\sigma}\mathfrak{F}_{k}(y,\sigma_{k})^{-1}D_{y}\mathfrak{F}_{k}(y,\sigma_{k})\dot{y}.

We obtain the conclusion from Lemma 4.4. ∎

Proposition 4.6.

For $k\gg 1$ , there is an open set $Y_{2r}\supset W_{k}\supset Y_{r}$ such that $(W_{k},\omega_{k},\Omega_{k})$ admits a equivariant special lagrangian fibration $f_{k}:W_{k}\longrightarrow B_{k}$ of phase $\theta_{k}$ over $B_{k}\subset\mathbb{R}^{n}$ , i.e. there is a $\Gamma$ -action on $B_{k}$ , $f_{k}$ is a $\Gamma$ -equivariant map, and $f_{k}$ is a special lagrangian fibration of phase $\theta_{k}$ , i.e.

\omega_{k}|_{f_{k}^{-1}(b)}\equiv 0,\ \ \ {\rm Im}e^{\sqrt{-1}\theta_{k}}\Omega_{k}|_{f_{k}^{-1}(b)}\equiv 0,

for any $b\in B_{k}$ .

Proof.

By Lemma 4.5, there is a unique $C^{1}$ -map

\sigma_{k}:B_{h_{E}}(0,\frac{3r}{2})\longrightarrow C^{1,\alpha}(d\Omega^{0}(T^{n})\oplus d^{*_{h}}\Omega^{2}(T^{n})),\ \ {\rm by}\ \ y\mapsto\sigma_{k}(y),

which satisfies

\mathfrak{F}_{k}(y,\sigma_{k}(y))=0,\ \ \ \|\sigma_{k}(y)\|_{C^{1,\alpha}(L,h)}\leq\delta\ll 1,

{\rm and}\ \ \ \|D\sigma_{k}(y)\|\leq 2n\delta\overline{C}C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)}.

This implies

|\frac{\partial\sigma_{k,j}(y)}{\partial y_{i}}|\leq 2n\delta\overline{C}C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)}\ll 1,

for $k\gg k_{1}>1$ .

Define a map $\Psi_{k}:Y_{\frac{3}{2}r}\longrightarrow Y_{2r}$ by

\Psi_{k}:(x,y)\mapsto(x,y_{1}+\sigma_{k,1}(y),\cdots,y_{n}+\sigma_{k,n}(y))=(x,y+\sigma_{k}(y)).

Note that the frame field $dx_{1},\cdots,dx_{n}$ induces local coordinates $x_{1},\cdots,x_{n}$ around any point on $L$ , and the differential can be expressed as

d\Psi_{k}:(\dot{x},\dot{y})\mapsto(\dot{x}_{j}+\sum\frac{\partial\sigma_{k,j}(y)}{\partial x_{i}}\dot{x}_{i},\dot{y}_{j}+\sum\frac{\partial\sigma_{k,j}(y)}{\partial y_{i}}\dot{y}_{i})

under such local coordinates. Thus $d\Psi_{k}$ is an isomorphism when $k\gg 1$ , which implies that $\Psi_{k}$ is an immersion. Furthermore, for $y_{1}\neq y_{2}\in\mathbb{R}^{n}$ ,

	$\displaystyle\Psi_{k}(x,y_{2})-\Psi_{k}(x,y_{1})$	$\displaystyle=$	$\displaystyle(x,\cdots,\int_{0}^{1}(1+\frac{\partial\sigma_{k,j}((1-t)y_{2}+ty_{1})}{\partial y_{j}}dt)(y_{2,j}-y_{1,j}),\cdots)$
		$\displaystyle\neq$	$\displaystyle 0.$

Hence $\Psi_{k}$ is an embedding.

Note that the $\Gamma$ -action on $Y_{2r}=T^{n}\times B_{h_{E}}(0,2r)$ preserves $\omega_{k},g_{k},\Omega_{k},\omega,g,\Omega$ , and is a product action on $T^{n}\times B_{h_{E}}(0,2r)$ , i.e. there are $\Gamma$ -actions on $T^{n}$ and $B_{h_{E}}(0,2r)$ such that $\gamma\cdot(x,y)=(\gamma\cdot x,\gamma\cdot y)$ for any $\gamma\in\Gamma$ , $x\in T^{n}$ , and $y\in B_{h_{E}}(0,2r)$ . Under the identification map (1),

	$\displaystyle(\gamma\cdot x,\gamma\cdot y)=(\gamma\cdot x,\iota(\gamma_{*}\sum_{j}y_{j}\frac{\partial}{\partial y_{j}})\omega)$	$\displaystyle=$	$\displaystyle(\gamma\cdot x,\gamma^{}\omega(\sum_{j}y_{j}\frac{\partial}{\partial y_{j}},\gamma^{-1}_{}\cdot))$
		$\displaystyle=$	$\displaystyle(\gamma\cdot x,\gamma^{-1,*}\sum_{j}y_{j}dx_{j}).$

Thus

$\displaystyle\gamma\cdot L(y,\sigma_{k}(y))$	$\displaystyle=$	$\displaystyle\{(\gamma\cdot x,\gamma\cdot(y_{1}+\sigma_{k,1}(y)(x),\cdots,y_{n}+\sigma_{k,n}(y)(x)))\|x\in T^{n}\}$
	$\displaystyle=$	$\displaystyle\{(\gamma\cdot x,\gamma^{-1,*}\sum_{j}(y_{j}+\sigma_{k,j}(y)(x))dx_{j}\|x\in T^{n}\}$
	$\displaystyle=$	$\displaystyle L(\gamma\cdot y,\gamma^{-1,*}\sigma_{k}(y)),$

for any $\gamma\in\Gamma$ and $y\in B_{h_{E}}(0,2r)$ . Since the $\Gamma$ -action preserves $\omega_{k}$ and $\Omega_{k}$ , $L(\gamma\cdot y,\gamma^{-1,*}\sigma_{k}(y))$ are special lagrangian submanifolds. By the uniqueness of $\sigma_{k}(y)$ , $\gamma^{-1,*}\sigma_{k}(y)=\sigma_{k}(\gamma\cdot y)\in C^{1,\alpha}(d\Omega^{0}(T^{n})\oplus d^{*_{h}}\Omega^{2}(T^{n}))$ . Hence

$\displaystyle\Psi_{k}(\gamma\cdot x,\gamma\cdot y)$	$\displaystyle=$	$\displaystyle(\gamma\cdot x,\gamma\cdot y+\sigma_{k}(\gamma\cdot y))$
	$\displaystyle=$	$\displaystyle(\gamma\cdot x,\gamma^{-1,*}\sum_{j}(y_{j}+\sigma_{k,j}(y))dx_{j})$
	$\displaystyle=$	$\displaystyle(\gamma\cdot x,\gamma\cdot(y_{1}+\sigma_{k,1}(y),\cdots,y_{n}+\sigma_{k,n}(y)))$
	$\displaystyle=$	$\displaystyle\gamma\cdot\Psi_{k}(x,y),$

i.e. $\Psi_{k}$ is a $\Gamma$ -equivariant map.

We denote $\mathcal{P}:Y_{2r}\longrightarrow B_{h_{E}}(0,2r)$ the natural projection, $B_{k}=B_{h_{E}}(0,\frac{3}{2}r)$ and $W_{k}=\Psi_{k}(Y_{\frac{3}{2}r})$ . Since the $\Gamma$ -action on $B_{h_{E}}(0,2r)$ preserves the metric $h_{E}$ and $0$ , $B_{k}=B_{h_{E}}(0,\frac{3}{2}r)$ is invariant. By $\delta\ll 1\ll r$ , $W_{k}\supset Y_{r}$ . Then $f_{k}=\mathcal{P}\circ\Psi_{k}^{-1}:W_{k}\longrightarrow B_{k}$ is a $\Gamma$ -equivariant special lagrangian fibration of $(W_{k},\omega_{k},\Omega_{k})$ of phase $\theta_{k}$ . We obtain the conclusion. ∎

	$\displaystyle\\|D_{y}\mathfrak{F}_{k}(y,\sigma_{k})\\|$	$\displaystyle\leq$	$\displaystyle C\\|(d\alpha_{k},d\beta_{k})\\|_{C^{1,\alpha}(Y_{2r},g)}(\sum_{l=0,1,\cdots,n}\\|\sigma_{k}\\|_{C^{1,\alpha}(L,h)}^{l})$
		$\displaystyle\leq$	$\displaystyle C\\|(d\alpha_{k},d\beta_{k})\\|_{C^{1,\alpha}(Y_{2r},g)}n\delta,$