Collapsing of Calabi-Yau manifolds and special lagrangian submanifolds

Yuguang Zhang Address: Department of Mathematics, Capital Normal University, Beijing, P.R.China Email address: yuguangzhang76@yahoo.com

Abstract.

In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite direction, it is shown that the existence of special lagrangian submanifolds with small volume implies the collapsing of some regions in the ambient Calabi-Yau manifolds.

[05D4]

1. Introduction

The notion of special lagrangian submanifold was introduced by Harvey and Lawson in the seminar paper [21]. Mclean studied the deformation theory of special lagrangian submanifolds in [28]. In the pioneer work [37], Stominger, Yau and Zaslow propose a conjecture about constructing the mirror manifold of a given Calabi-Yau manifold, the SYZ conjecture, via special lagrangian fibrations. Since then, lots of works were devoted to study special lagrangian submanifolds and fibrations (c.f. [22], [31], [32], [33], [15], [16], [27], [17], [36], [39], [24], [25], and references in [25]). In [26] and [19], a refined version of SYZ conjecture was proposed by using the collapsing of Ricci-flat Calabi-Yau manifolds in the Gromov-Hausdorff sense. These two versions of SYZ conjecture suggest a relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds. In this paper, we study this relationship.

If $(M,\omega,J,g)$ is a compact Ricci-flat Kähler $n$ -manifold, and admits a no-where vanishing holomorphic $n$ -form $\Omega$ , the holomorphic volume form, $(M,\omega,J,g,\Omega)$ is called a Ricci-flat Calabi-Yau $n$ -manifold, and $(\omega,J,g,\Omega)$ is called a Calabi-Yau structure on $M$ . We can normalize $\Omega$ such that

\frac{\omega^{n}}{n!}=\frac{(-1)^{\frac{n^{2}}{2}}}{2^{n}}\Omega\wedge\overline{\Omega},

(c.f. [25]). Yau’s theorem of Calabi conjecture guarantees the existence of Ricci-flat Kähler metrics on Kähler manifolds with trivial canonical bundle (c.f. [40]), which implies the existence of Calabi-Yau structures on such manifolds. The holonomy group of a Ricci-flat Calabi-Yau $n$ -manifold is a subgroup of $SU(n)$ . The study of Calabi-Yau manifolds is important in both mathematics and physics (c.f. [41]).

A special lagrangian submanifold $L$ of phase $\theta\in\mathbb{R}$ in a Ricci-flat Calabi-Yau $n$ -manifold $(M,\omega,J,g,\Omega)$ is a lagrangian submanifold $L\subset M$ corresponding to the Kähler form $\omega$ such that ${\rm Re}e^{\sqrt{-1}\theta}\Omega|_{L}=dv_{g|_{L}}$ where $dv_{g|_{L}}$ denotes the volume form of $g|_{L}$ on $L$ . Equivalently, $\dim_{\mathbb{R}}L=n$ ,

\omega|_{L}\equiv 0,\ \ \ {\rm Im}e^{\sqrt{-1}\theta}\Omega|_{L}\equiv 0

(c.f. [21]). In [28], Mclean showed that, for a compact special lagrangian submanifold $L$ in a Calabi-Yau manifold $(M,\omega,J,g,\Omega)$ , the local moduli space of special lagrangian submanifolds near $L$ is a smooth manifold of dimension $b_{1}(L)$ , and, moreover, the tangent space of the moduli space at $L$ can be identified with the space of harmonic 1-forms on $(L,g|_{L})$ . In [22], various structures on the moduli space of special lagrangian submanifolds were studied.

A special lagrangian fibration on a Calabi-Yau $n$ -manifold $(M,\omega,\Omega)$ consists of a topological space $B$ , and a surjection $f:M\longrightarrow B$ such that there is an open dense subset $B_{0}\subset B$ , which is a real $n$ -manifold, satisfying that, for any $b\in B_{0}$ , $f^{-1}(b)$ is a smooth special lagrangian submanifold in $(M,\omega,\Omega)$ . By [10] (see also [18]), $f^{-1}(b)$ , $b\in B_{0}$ , is a $n$ -torus. The first step of SYZ conjecture is to construct such fibration on a Calabi-Yau manifold when the complex structure is close to the large complex structure limit point enough (c.f. [37]). Then the mirror manifold is a compactification of the dual fibration of $f:f^{-1}(B_{0})\longrightarrow B_{0}$ . Generalized special lagrangian fibrations were constructed in some almost Calabi-Yau manifolds in [31], [32], [33], [16]. In [34], H-minimal Lagrangian fibrations, a generalization of special lagrangian fibration, were constructed on some regions of Kähler-Einstein manifolds with negative scalar curvature.

In [26] and [19], SYZ conjecture was refined to the following form: Let $\pi:\mathcal{M}\rightarrow\Delta$ be a maximally unipotent degeneration of Calabi-Yau $n$ -manifolds over the unit disc $\Delta\subset\mathbb{C}$ , and $\alpha$ be an ample class on $\mathcal{M}$ . For any $t\in\Delta\backslash\{0\}$ , let $\tilde{g}_{t}$ be the unique Ricci-flat Kähler metric on $M_{t}=\pi^{-1}(t)$ with its Kähler form $\tilde{\omega}_{t}\in\alpha|_{M_{t}}\in H^{1,1}(M_{t},\mathbb{R})$ , and $\bar{g}_{t}={\rm diam}_{\tilde{g}_{t}}^{-2}(M)\tilde{g}_{t}$ . Then $(M_{t},\bar{g}_{t})$ converges to a compact metric space $(B,d_{B})$ of Hausdorff dimension $n$ in the Gromov-Hausdorff sense, when $t\rightarrow 0$ . This conjecture was verified for some K3 surfaces in [19]. The two versions of SYZ conjecture suggest the equivalence between the existence of special lagrangian submanifolds and the collapsing of Ricci-flat Kähler metrics on some regions of Calabi-Yau manifolds, when complex structures are close to the large complex limit point enough.

In Riemannian geometry, the collapsing of Riemannian manifolds was studied by various authors (c.f. [5], [6], [4], [8], [11], and references in [11]), since Gromov introduced the notion of Gromov-Hausdorff topology in [14]. In [6], it was proved that there is a constant $\epsilon_{0}(n)>0$ depending only on $n$ such that there is an $F$ -structure of positive rank on the region $M_{\epsilon_{0}}$ in a Riemannian $n$ -manifold $(M,g)$ , where $M_{\epsilon_{0}}$ denotes the subset with injectivity radius $i_{g}(p)<\epsilon_{0}$ and sectional curvature $\sup\limits_{B_{g}(p,1)}|K_{g}|\leq 1,$ for any $p\in M_{\epsilon_{0}}$ . See [5] and [6] for the definition of $F$ -structure of positive rank, which is a generalization of fibration. A folklore conjecture says that there should be special lagrangian fibrations on such region in a Calabi-Yau manifold, i.e. the region of bounded curvature and sufficiently collapsed (c.f. [12]). The first result in the present paper is devoted to construct special lagrangian fibrations under such Riemannian geometric conditions.

[05D5]

Theorem 1.1.

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.

[05D6]

Remark 1.2.

From the proof of this theorem, we can see that $B$ is an orbifold, and, if $b$ belongs the singular set of $B$ , $f^{-1}(b)$ is a smooth multi-fiber.

[05D7]

Remark 1.3.

The condition ii) in the theorem can be replaced by the following small non-vanishing $n$ -cycle condition: there is an $[A]\in H_{n}(B_{g}(p,\sigma i_{g}(p)),\mathbb{Z})$ such that

\int_{A}\Omega\neq 0.

This condition can not be removed since it is satisfied if there is a special lagrangian submanifold $L$ near $p$ having comparable size to $i_{g}(p)$ , for example $L\subset B_{g}(p,\sigma i_{g}(p))$ .

[05D8]

Remark 1.4.

It is a challenging task to verify condition i) in Theorem 1.1, i.e. to find the region of bounded curvature in a Ricci-flat Calabi-Yau manifold. If $(M,\omega,J,g,\Omega)$ is a K3-surface with Ricci-flat metric, it was shown in [8] that there are universal constants $C>0$ , $\tau>0$ , and a finite subset $\{p_{j}\}\subset M$ , $1\leq j\leq\tau$ , such that

\sup_{B_{g}(p,1)}|K_{g}|\leq C,

for any $p\in M\backslash\bigcup\limits_{1\leq j\leq\tau}B_{g}(p_{j},2)$ . From the author’s knowledge, no such estimate for higher dimensional Calabi-Yau manifolds is known except some trivial cases, for example $K3\times T^{2}$ .

Next, in the opposite direction, we show that the existence of special lagrangian submanifolds with small volume implies the collapsing of some regions in the ambient Calabi-Yau manifolds. The following theorem is a corollary of a volume comparison theorem for calibrated submanifolds in [16].

[05D9]

Theorem 1.5.

Let $(M,\omega,J,g,\Omega)$ be a closed Ricci-flat Calabi-Yau n-manifold, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{B_{g}(p,2\pi)}K_{g}\leq 1,

and there is a special lagrangian submanifold $L$ of phase $\theta$ such that $p\in L$ , and

\int_{L}{\rm Re}e^{\sqrt{-1}\theta}\Omega<\frac{\pi}{2n}\varpi_{n-1},

where $\varpi_{n-1}$ denotes the volume of $S^{n-1}$ with the standard metric of constant curvature 1. Then the injectivity radius $i_{g}(p)$ of $(M,g)$ at $p$ satisfies that

i_{g}(p)^{n}\leq\frac{n\pi^{n-1}}{2^{n-1}\varpi_{n-1}}\int_{L}{\rm Re}e^{\sqrt{-1}\theta}\Omega.

Let $\{(M_{k},\omega_{k},J_{k},g_{k},\Omega_{k})\}$ be a family of closed Ricci-flat Calabi-Yau $n$ -manifolds, $L_{k}\subset M_{k}$ be special lagrangian submanifolds of phase $\theta_{k}$ such that

\lim_{k\longrightarrow\infty}\int_{L_{k}}{\rm Re}e^{\sqrt{-1}\theta_{k}}\Omega_{k}=0,

and $\{p_{k}\}$ be a sequence of points satisfying that $p_{k}\in L_{k}$ , and $\sup\limits_{B_{g_{k}}(p_{k},2\pi)}K_{g_{k}}\leq 1$ . The above theorem implies that

\lim_{k\longrightarrow\infty}i_{g_{k}}(p_{k})=0,

and, by passing to a subsequence, $\{(M_{k},g_{k},p_{k})\}$ converges to a path metric space of lower dimension in the pointed Gromov-Hausdorff sense (c.f. [8], [3]). Theorem 1.1 and Theorem 1.5 give an evidence of the equivalence between the existence of special lagrangian submanifolds and the collapsing of Ricci-flat Kähler metrics on Calabi-Yau manifolds near the large complex limit point from the Riemannian geometry’s point of view.

The organization of the paper is as follows: In §2, we review some notions and results, which will be used in this paper. In §3, we use the blow-up argument to give local approximations of Calabi-Yau manifolds by complete flat Calabi-Yau manifolds. In §4, we study the deformation of special lagrangian fibrations. In §5, we prove Theorem 1.1 by combining the results in §3 and §4. Finally, we prove Theorem 1.5 in §6.

Acknowledgement: The author would like to thank Prof. Weidong Ruan and Prof. Xiaochun Rong for useful discussions. Thanks also goes to Prof. Fuquan Fang for constantly support.

[05DA]

2. Preliminaries

In this section, we review some notions and results, which will be used in the proof of Theorem 1.1.

[05DB]

2.1. Cheeger-Gromov convergence

Since Gromov introduced the concept of Gromov-Hausdorff topology in [14], the convergence of Riemannian manifolds was studied from various perspectives (c.f. [1], [2], [7], [8], [11], [13], [19], [35], [38] and references in [9]). In [14] and [13], a convergence theorem, the Cheeger-Gromov convergence theorem, was proved for Riemannian manifolds with bounded curvature and non-collapsing. The Kähler version of this theorem can be found in [30]. See [7] for the convergence of manifolds with other holonomy groups.

[05DC]

Theorem 2.1 (Kähler version of Cheeger-Gromov convergence theorem).

Let $\{(M_{k},g_{k},J_{k},\omega_{k},p_{k})\}$ be a family of pointed compact Kähler n-manifolds with sectional curvature and injectivity radius at $p_{k}$

|K_{g_{k}}|\leq 1,\ \ \ i_{g_{k}}(p_{k})\geq C,

for a constant $C>0$ independent of $k$ . Then a subsequence of $\{(M_{k},g_{k},J_{k},\omega_{k},p_{k})\}$ converges to a complete Kähler n-manifold $(X,g,J,\omega,p)$ in the pointed $C^{1,\alpha}$ -sense, i.e. for any $r>0$ , there are embeddings $F_{k,r}:B_{g}(p,r)\longrightarrow M_{k}$ such that $F_{k,r}(p)=p_{k}$ , $F_{k,r}^{*}g_{k}$ (resp. $dF_{k,r}^{-1}J_{k}dF_{k,r}$ and $F_{k,r}^{*}\omega_{k}$ ) converges to $g$ (resp. $J$ and $\omega$ ) in the $C^{1,\alpha}$ -sense.

If we assume that $g_{k}$ are Einstein metrics, it is shown in [1] that, by passing to a subsequence, $\{(M_{k},g_{k},J_{k},\omega_{k},p_{k})\}$ converges to $(X,g,J,\omega,p)$ in the pointed $C^{\infty}$ -sense, and $g$ is also an Einstein metric, i.e. $F_{k,r}^{*}g_{k}$ (resp. $dF_{k,r}^{-1}J_{k}dF_{k,r}$ and $F_{k,r}^{*}\omega_{k}$ ) converges to $g$ (resp. $J$ and $\omega$ ) in the $C^{\infty}$ -sense. Assume that $(M_{k},g_{k},J_{k},\omega_{k},p_{k})$ are Ricci-flat Calabi-Yau manifolds, and $\Omega_{k}$ are the corresponding holomorphic volume forms. Since $\Omega_{k}$ are parallel, i.e. $\nabla^{g_{k}}\Omega_{k}\equiv 0$ , for any $r>0$ , $F_{k,r}^{*}\Omega_{k}$ converge to a holomorphic volume form $\Omega$ on $X$ in the $C^{\infty}$ -sense, and $(X,g,J,\omega,\Omega)$ is a complete Ricci-flat Calabi-Yau $n$ -manifold.

In [5], [6], the collapsing of Riemannian manifolds with bounded curvature was studied by combining blow-up arguments and the Cheeger-Gromov convergence theorem. It was shown that there is a constant $\epsilon_{0}(n)>0$ depending only on $n$ such that there is an $F$ -structure $\mathcal{F}$ of positive rank on a region covering $M_{\epsilon_{0}}$ in a Riemannian $n$ -manifold $(M,g)$ , where $M_{\epsilon_{0}}$ denotes the subset with injectivity radius $i_{g}(p)<\epsilon_{0}$ and sectional curvature $\sup\limits_{B_{g}(p,1)}|K_{g}|\leq 1,$ for any $p\in M_{\epsilon_{0}}$ . See [5] and [6] for the definition of $F$ -structure of positive rank. If we assume that $g$ is a Kähler metric, some additional information about the $F$ -structure $\mathcal{F}$ is expected. We have the following conjecture:

[05DD]

Conjecture 2.2.

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.

We will address this question in other papers. In the present paper, we prove Theorem 1.1 by combining Theorem 2.1 and the deformation theory of special lagrangian fibrations.

[05DE]

2.2. Implicit function theorem

For studying the deformation of special lagrangian fibrations, we need the following quantity version of implicit function theorem.

[05DF]

Theorem 2.3 (Theorem 3.2 in [31]).

Let $(\mathfrak{B}_{1},\|\cdot\|_{1})$ and $(\mathfrak{B}_{2},\|\cdot\|_{2})$ be two Banach spaces, $\|\cdot\|_{E}$ be the standard Euclidean metric on $\mathbb{R}^{n}$ , $U\subset\mathbb{R}^{n}\times\mathfrak{B}_{1}$ be an open set, and $\mathfrak{F}:U\longrightarrow\mathfrak{B}_{2}$ be a continuously differentiable map. Denote the differential

D\mathfrak{F}(y,\sigma)(\dot{y}+\dot{\sigma})=D_{y}\mathfrak{F}(y,\sigma)\dot{y}+D_{\sigma}\mathfrak{F}(y,\sigma)\dot{\sigma},

for $(y,\sigma)\in U$ , $\dot{y}\in\mathbb{R}^{n}$ and $\dot{\sigma}\in\mathfrak{B}_{1}$ . Assume that $(0,0)\in U$ satisfies that $D_{\sigma}\mathfrak{F}(0,0):\mathfrak{B}_{1}\longrightarrow\mathfrak{B}_{2}$ has a bounded linear inverse $D_{\sigma}\mathfrak{F}(0,0)^{-1}:\mathfrak{B}_{2}\longrightarrow\mathfrak{B}_{1}$ with

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

for a constant $\overline{C}>0$ . Let $r>0$ , $\delta_{0}>\delta>0$ be constants such that, if $\|y_{0}\|_{E}<r,$ and $\|\sigma\|_{1}\leq\delta_{0}$ , then $(y_{0},\sigma)\in U$ , and

\|D_{\sigma}\mathfrak{F}(y_{0},\sigma)-D_{\sigma}\mathfrak{F}(0,0)\|\leq\frac{1}{2\overline{C}}\ \ {\rm and}\ \ \|\mathfrak{F}(y_{0},0)\|_{2}\leq\frac{\delta}{4\overline{C}}.

Then, for any $\|y\|_{E}<r$ , there exists a unique $\sigma(y)\in\mathfrak{B}_{1}$ such that

\mathfrak{F}(y,\sigma(y))=0,\ \ \ \|\sigma(y)\|_{1}\leq\delta.

Furthermore,

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

The difference between this version of implicit function theorem and the usual one (c.f. [20]) is that we use the condition $\|\mathfrak{F}(y,0)\|_{2}\leq\frac{\delta}{4\overline{C}}$ to replace the condition $\mathfrak{F}(0,0)=0$ besides other quantity estimates.

[05DG]

3. The blow-up limit

Let $\{(M_{k},\omega_{k},J_{k},g_{k},\Omega_{k})\}$ be a family of closed Ricci-flat Calabi-Yau $n$ -manifold with $[\omega_{k}]\in H^{2}(M_{k},\mathbb{Z})$ , and $p_{k}\in M_{k}$ . Assume 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)

there is a $\sigma\gg 1$ such that $[\Omega_{k}|_{B_{g_{k}}(p_{k},\sigma i_{g_{k}}(p_{k}))}]\neq 0$ in $H^{n}(B_{g_{k}}(x_{k},\sigma i_{g_{k}}(p_{k})),\mathbb{C})$ .

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

i_{\tilde{g}_{k}}(p_{k})=1,\ \ \ \sup_{B_{\tilde{g}_{k}}(p_{k},k)}|K_{\tilde{g}_{k}}|\leq\frac{1}{k^{2}},

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. The purpose of this section is to prove that $(X,\omega_{0},\Omega_{0})$ admits a special lagrangian fibration.

By the smooth convergence, $i_{g_{0}}(p_{0})=\lim\limits_{k\rightarrow\infty}i_{\tilde{g}_{k}}(p_{k})=1$ . 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.

[05DH]

Lemma 3.1.

$\dim_{\mathbb{R}}S\geq n$ .

[05DI]

Proof.

If $\dim_{\mathbb{R}}S<n$ , then

H^{n}(X,\mathbb{C})=H^{n}(T_{r}(S),\mathbb{C})=H^{n}(S,\mathbb{C})=\{0\}

for any $r>0$ , where $T_{r}(S)=\{p\in X|dist_{g_{0}}(p,S)\leq r\}$ . Let $r_{0}>r_{1}>\sigma$ such that $T_{r_{1}}(S)\subset B_{g_{0}}(x_{0},r_{0})$ , and $F_{r_{0},k}(T_{r_{1}}(S))\supset B_{\tilde{g}_{k}}(x_{k},\sigma)$ for $k\gg 1$ . Then the inclusion maps induce homeomorphisms on cohomology groups

H^{n}(M,\mathbb{C})\longrightarrow H^{n}(F_{r_{0},k}(T_{r_{1}}(S)),\mathbb{C})\longrightarrow H^{n}(B_{\tilde{g}_{k}}(x_{k},\sigma),\mathbb{C}),

{\rm and}\ \ \ [\tilde{\Omega}_{k}]\mapsto[\tilde{\Omega}_{k}|_{F_{r_{0},k}(T_{r_{1}}(S))}]\mapsto[\tilde{\Omega}_{k}|_{B_{\tilde{g}_{k}}(x_{k},\sigma)}]\neq 0.

Thus $[\tilde{\Omega}_{k}|_{F_{r_{0},k}(T_{r_{1}}(S))}]\neq 0$ in $H^{n}(F_{r_{0},k}(T_{r_{1}}(S)),\mathbb{C})$ , which contradicts to

H^{n}(F_{r_{0},k}(T_{r_{1}}(S)),\mathbb{C})\cong H^{n}(T_{r_{1}}(S),\mathbb{C})=\{0\}.

∎

If $\pi_{h}:\tilde{S}\longrightarrow S$ is the holonomy covering of $S$ , Bieberbach’s theorem (c.f. [6], [29]) says that $(\tilde{S},\pi_{h}^{*}g_{0})$ is isometric to a flat torus, and $\pi_{h}$ has finite order at most $\lambda(n)$ , for a constant $\lambda(n)$ depending only on $n$ . If we denote $\bar{\pi}:\mathbb{C}^{n}\longrightarrow X$ the universal covering of $X$ with $\bar{\pi}(0)\in S$ , then $\bar{S}=\bar{\pi}^{-1}(S)$ is a real linear subspace of $\mathbb{C}^{n}$ , and $\omega_{E}=\bar{\pi}^{*}\omega_{0}$ (resp. $\Omega_{E}=\bar{\pi}^{*}\Omega_{0}$ ) is the standard flat Kähler form (resp. the standard holomorphic volume form), i.e. $\omega_{E}=\sqrt{-1}\sum_{\alpha}dz_{\alpha}\wedge d\bar{z}_{\alpha}$ and $\Omega_{E}=dz_{1}\wedge\cdots\wedge dz_{n}$ under some coordinates $z_{1},\cdots,z_{n}$ on $\mathbb{C}^{n}$ . Note that there is a lattice $\Lambda\subset\bar{S}$ such that $\tilde{S}=\bar{S}/\Lambda$ . If we denote $\mathfrak{q}:\bar{S}\longrightarrow\tilde{S}$ the quotient map, then $\bar{\pi}=\pi_{h}\circ\mathfrak{q}$ .

[05DJ]

Lemma 3.2.

$\dim_{\mathbb{R}}\bar{S}=n$ , and there is a constant $\theta_{0}\in\mathbb{R}$ such that $\omega_{E}|_{\bar{S}}=0$ and ${\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{E}|_{\bar{S}}=0.$ Moreover, $S$ is a special lagrangian submanifold of phase $\theta_{0}$ in $(X,\omega_{0},\Omega_{0})$ , i.e. $\dim_{\mathbb{R}}S=n$ ,

\omega_{0}|_{S}\equiv 0,\ \ {\rm and}\ \ {\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{0}|_{S}=0.

[05DK]

Proof.

If $\omega_{E}|_{\bar{S}}\neq 0$ , and thus $\omega_{0}|_{S}\neq 0$ , then there are two vectors $v_{1},v_{2}\in\bar{S}$ such that $\omega_{E}(v_{1},v_{2})>0$ . By perturbing $v_{1}$ and $v_{2}$ a little bit if necessary, we have that $\tilde{\Sigma}=\mathfrak{q}(\{t_{1}v_{1}+t_{2}v_{2}|t_{i}\in\mathbb{R}\})$ is a closed 2-torus in $\tilde{S}$ , i.e. a closed 2-parameters subgroup. Thus $\Sigma=\pi_{h}(\tilde{\Sigma})$ is a closed oriented surface in $S$ , which satisfies

\int_{\Sigma}\omega_{0}\geq\frac{1}{\lambda(n)}\int_{\tilde{\Sigma}}\pi_{h}^{*}\omega_{0}\geq\frac{\omega_{E}(v_{1},v_{2})}{\lambda(n)\|v_{1}\wedge v_{2}\|_{h_{E}}}V_{E}>0,

where $V_{E}$ denotes the Euclidean area of the intersection of $\{t_{1}v_{1}+t_{2}v_{2}|t_{i}\in\mathbb{R}\}$ with the fundamental domain of the quotient map $\mathfrak{q}$ . From the smooth convergence of $(M_{k},\tilde{\omega}_{k},\tilde{g}_{k})$ ,

\lim_{k\longrightarrow\infty}i^{-2}_{g_{k}}(p_{k})\int_{F_{r,k}(\Sigma)}\omega_{k}=\lim_{k\longrightarrow\infty}\int_{F_{r,k}(\Sigma)}\tilde{\omega}_{k}=\lim_{k\longrightarrow\infty}\int_{\Sigma}F_{r,k}^{*}\tilde{\omega}_{k}=\int_{\Sigma}\omega_{0},

for $r\gg 1$ such that $\Sigma\subset B_{g_{0}}(p_{0},r)$ . Thus

0<\frac{1}{2}i^{2}_{g_{k}}(p_{k})\int_{\Sigma}\omega_{0}\leq\int_{F_{r,k}(\Sigma)}\omega_{k}\leq 2i^{2}_{g_{k}}(p_{k})\int_{\Sigma}\omega_{0}\leq 2k^{-2}\int_{\Sigma}\omega_{0}<1,

for $k\gg 1$ . Since $[F_{r,k}(\Sigma)]\in H_{2}(M_{k},\mathbb{Z})$ and $[\omega_{k}]\in H^{2}(M_{k},\mathbb{Z})$ , we obtain

\int_{F_{r,k}(\Sigma)}\omega_{k}\in\mathbb{Z},

which is a contradiction. Hence $\omega_{E}|_{\bar{S}}\equiv 0$ and $\omega_{0}|_{S}\equiv 0$ , which implies that $S$ is a lagrangian submanifold $(X,\omega_{0})$ by combining Lemma 3.1.

Since $\bar{S}$ is a lagrangian linear subspace of $(\mathbb{C}^{n},\omega_{E})$ , there is a $\theta_{0}\in\mathbb{R}$ such that ${\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{E}|_{\bar{S}}=0$ . This implies that $\bar{S}$ is a special lagrangian linear subspace of phase $\theta_{0}$ in $(\mathbb{C}^{n},\omega_{E},\Omega_{E})$ . Thus $S$ is a special lagrangian submanifold of phase $\theta_{0}$ in $(X,\omega_{0},\Omega_{0})$ , i.e.

\omega_{0}|_{S}=0,\ \ \ {\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{0}|_{S}=0.

∎

[05DL]

Lemma 3.3.

For $k\gg 1$ , $[F_{r,k}^{*}\tilde{\omega}_{k}|_{S}]=0$ in $H^{2}(S,\mathbb{R})$ .

[05DM]

Proof.

By the smooth convergence of $\tilde{\omega}_{k}$ and Lemma 3.2,

\lim_{k\longrightarrow\infty}\int_{F_{r,k}(A)}\tilde{\omega}_{k}=\int_{A}\omega_{0}=0,

for any cycle $A\in H_{2}(S,\mathbb{Z})$ . For $k\gg 1$ , we have

|\int_{F_{r,k}(A)}\omega_{k}|=i^{2}_{g_{k}}(p_{k})|\int_{F_{r,k}(A)}\tilde{\omega}_{k}|<\frac{1}{2k^{2}}<1.

Since $[F_{r,k}(A)]\in H_{2}(M_{k},\mathbb{Z})$ and $[\omega_{k}]\in H^{2}(M_{k},\mathbb{Z})$ , we obtain $\int_{F_{r,k}(A)}\omega_{k}\in\mathbb{Z}$ . This implies that $|\int_{F_{r,k}(A)}\omega_{k}|=0$ , and we obtain the conclusion

\int_{A}F_{r,k}^{*}\tilde{\omega}_{k}=0.

∎

Let $\tilde{X}$ be the total space of the pull-back $\pi_{h}^{*}\nu(S)$ of the normal bundle. Note that we can identify the zero section of $\pi_{h}^{*}\nu(S)$ with $\tilde{S}$ , and the covering $\pi_{h}$ extends to a finite covering $\pi:\tilde{X}\longrightarrow X$ of $X$ , i.e. $\tilde{S}=\pi^{-1}(S)\subset\tilde{X}$ , and $\pi|_{\tilde{S}}=\pi_{h}$ . The fundamental group $\pi_{1}(\tilde{S})\cong\pi_{1}(\tilde{X})$ is isomorphic to the lattice $\Lambda$ , $\pi_{1}(\tilde{X})$ is a normal subgroup of $\pi_{1}(X)=\pi_{1}(S)$ , and the covering group $\Gamma\cong\pi_{1}(S)/\pi_{1}(\tilde{S})=\pi_{1}(X)/\pi_{1}(\tilde{X})$ . Note that $\pi_{1}(\tilde{X})$ (resp. $\pi_{1}(X)$ ) acts on $\mathbb{C}^{n}$ preserving $g_{E}$ , $\omega_{E}$ and $\Omega_{E}$ , $\bar{S}$ is invariant, $\tilde{X}=\mathbb{C}^{n}/\pi_{1}(\tilde{X})$ (resp. $X=\mathbb{C}^{n}/\pi_{1}(X)$ ), and $\tilde{S}=\bar{S}/\pi_{1}(\tilde{S})=\bar{S}/\Lambda$ (resp. $S=\bar{S}/\pi_{1}(S)$ ).

[05DN]

Proposition 3.4.

Let $\bar{S}^{\perp}$ be the orthogonal complement of $\bar{S}$ in $\mathbb{C}^{n}$ , i.e. $\mathbb{C}^{n}=\bar{S}\oplus\bar{S}^{\perp}$ , and $g_{E}(v,w)=0$ , for any $v\in\bar{S}$ and $w\in\bar{S}^{\perp}$ . Then

i)

$(\tilde{X},\pi^{*}g_{0})$ is isometric to $(T^{n}\times\bar{S}^{\perp},h+h_{E})$ , where $T^{n}=\bar{S}/\Lambda=\tilde{S}$ , $h_{E}=g_{E}|_{\bar{S}^{\perp}}$ , and $h$ is the standard flat metric on $T^{n}$ induced by $g_{E}|_{\bar{S}}$ .
ii)

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

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 $y\in\bar{S}^{\perp}$ , and a constant $\theta_{0}\in\mathbb{R}$ .

[05DP]

Proof.

We choose coordinates $x_{1},\cdots,x_{n}$ on $\bar{S}$ and $y_{1},\cdots,y_{n}$ on $\bar{S}^{\perp}$ such that

g_{E}=\sum(dx_{j}^{2}+dy_{j}^{2}),\ \ \ \omega_{E}=\sum dx_{j}\wedge dy_{j},\ \ \ e^{\sqrt{-1}\theta_{0}}\Omega_{E}=\bigwedge_{j=1}^{n}(dx_{j}+\sqrt{-1}dy_{j}).

If $\mathcal{G}$ is a subgroup of the fundamental group $\pi_{1}(X)=\pi_{1}(S)$ , then $\mathcal{G}$ acts on $\mathbb{C}^{n}$ preserving $g_{E}$ , $\omega_{E}$ and $\Omega_{E}$ , and $\bar{S}$ is a invariant subspace. For any $\gamma\in\mathcal{G}$ , we have $\gamma\cdot(v+w)=G_{\gamma}(v+w)+b_{\gamma}$ , where $G_{\gamma}\in U(\mathbb{C}^{n})$ , $b_{\gamma}\in\bar{S}$ , $v\in\bar{S}$ and $w\in\bar{S}^{\perp}$ . Since $\bar{S}$ is invariant, we obtain then $G_{\gamma}(v+w)=A_{\gamma}v+B_{\gamma}w+C_{\gamma}w$ where $A_{\gamma}\in SO(\bar{S})$ , $B_{\gamma}\in SO(\bar{S}^{\perp})$ , and $C_{\gamma}\in Hom(\bar{S}^{\perp},\bar{S})$ . Moreover, $G_{\gamma}\in SO(\mathbb{R}^{2n})$ implies $C_{\gamma}=0$ . Since $\omega_{E}(G_{\gamma}(v+w),G_{\gamma}(v+w))=\omega_{E}(v+w,v+w)$ , we have $B_{\gamma}=A_{\gamma}^{-1,T}=A_{\gamma}$ , and $\gamma\cdot(v+w)=A_{\gamma}(v+w)+b_{\gamma}$ . Thus $\pi_{1}(\tilde{X})\cong\Lambda$ acts on $\mathbb{C}^{n}$ given by $\gamma\cdot(v+w)=v+w+b_{\gamma}$ , $b_{\gamma}\in\Lambda$ , for any $v\in\bar{S}$ and $w\in\bar{S}^{\perp}$ . This implies that $\tilde{X}=\mathbb{C}^{n}/\pi_{1}(\tilde{X})\cong\bar{S}/\Lambda\times\bar{S}^{\perp}=\tilde{S}\times\bar{S}^{\perp}$ , and $\pi^{*}g_{0}=h+h_{E}$ where $h_{E}=g_{E}|_{\bar{S}^{\perp}}$ , and $h$ is the standard flat metric on $\tilde{S}$ induced by $g_{E}|_{\bar{S}}$ .

The $\pi_{1}(X)$ -action on $\mathbb{C}^{n}$ descents to a $\Gamma$ -action on $\tilde{X}$ , which is a product action since the $\pi_{1}(X)$ -action is so. Moreover, $\tilde{S}\times\{0\}$ is a invariant set as $\bar{S}\times\{0\}$ is invariant under the $\pi_{1}(X)$ -action. If we denote the quotient map $\mathfrak{q}_{1}:\mathbb{C}^{n}\longrightarrow\mathbb{C}^{n}/\Lambda=\tilde{X}$ , then $\bar{\pi}=\pi\circ\mathfrak{q}_{1}$ , $g_{E}=\mathfrak{q}_{1}^{*}\pi^{*}g_{0}$ , $\omega_{E}=\mathfrak{q}_{1}^{*}\pi^{*}\omega_{0}$ , and $\Omega_{E}=\mathfrak{q}_{1}^{*}\pi^{*}\Omega_{0}$ . Since $\omega_{E}|_{\bar{S}\times\{y\}}=0$ and $e^{\sqrt{-1}\theta_{0}}\Omega_{E}|_{\bar{S}\times\{y\}}=0$ for $y\in\bar{S}^{\perp}$ , we obtain that

\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 a constant $\theta_{0}\in\mathbb{R}$ . ∎

[05DQ]

Remark 3.5.

The coordinates $x_{1},\cdots,x_{n}$ on $\bar{S}$ in the proof of this proposition induce parallel 1-forms $dx_{1},\cdots,dx_{n}$ on $(\tilde{S},h)$ £¬ which are pointwise linear independent, i.e. $dx_{1},\cdots,dx_{n}$ is a global parallel frame field. Under the coordinates $y_{1},\cdots,y_{n}$ on $\bar{S}^{\perp}$ , we have these formulas

\pi^{*}g_{0}=\sum(dx_{j}^{2}+dy_{j}^{2}),\ \ \ \pi^{*}\omega_{0}=\sum dx_{j}\wedge dy_{j},\ \ \ e^{\sqrt{-1}\theta_{0}}\pi^{*}\Omega_{0}=\bigwedge_{j=1}^{n}(dx_{j}+\sqrt{-1}dy_{j}).

[05DR]

Remark 3.6.

The natural projection $f_{0}:\tilde{X}\longrightarrow\bar{S}^{\perp}$ is equivariant under the $\Gamma$ actions on $\tilde{X}$ and $\bar{S}^{\perp}$ . For any $y\in\bar{S}^{\perp}$ , $f_{0}^{-1}(y)=\tilde{S}\times\{y\}$ , and $f_{0}$ is a special lagrangian fibration on $(\tilde{X},\pi^{*}\omega_{0},e^{\sqrt{-1}\theta_{0}}\pi^{*}\Omega_{0})$ , i.e. $\dim_{\mathbb{R}}f_{0}^{-1}(y)=n$ ,

\pi^{*}\omega_{0}|_{f_{0}^{-1}(y)}\equiv 0,\ \ \ e^{\sqrt{-1}\theta_{0}}\pi^{*}\Omega_{0}|_{f_{0}^{-1}(y)}\equiv 0.

[05DS]

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.

[05DT]

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 $\mathcal{U}(r)=\{\|y\|_{h_{E}}+\|\sigma\|_{C^{1,\alpha}(L,h)}<2r|(y,\sigma)\in\mathbb{R}^{n}\times\mathfrak{B}_{1}\}$ .

[05DU]

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 $k$ .

[05DV]

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}}.

[05DW]

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 $k$ .

[05DX]

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 $k$ . ∎

[05DY]

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}.

[05DZ]

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}.

∎

[05E0]

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 $k$ .

[05E1]

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. ∎

[05E2]

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 $b\in B_{k}$ .

[05E3]

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. ∎

[05E4]

5. Proof of Theorem 1.1

Now we are ready to prove Theorem 1.1.

[05E5]

Proof of Theorem 1.1.

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

i_{\tilde{g}_{k}}(p_{k})=1,\ \ \ \sup_{B_{\tilde{g}_{k}}(p_{k},k)}|K_{\tilde{g}_{k}}|\leq\frac{1}{k^{2}},

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

\pi^{*}g_{0}=\sum(dx_{j}^{2}+dy_{j}^{2}),\ \ \ \pi^{*}\omega_{0}=\sum dx_{j}\wedge dy_{j},\ \ \ e^{\sqrt{-1}\theta_{0}}\pi^{*}\Omega_{0}=\bigwedge_{j=1}^{n}(dx_{j}+\sqrt{-1}dy_{j}).

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

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

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. ∎

[05E6]

6. Estimates for injectivity radius

In [21], Harvey and Lawson introduced the notion of calibrated submanifold. If $(M,g)$ is a Riemannian manifold, and $\Theta$ is a closed $n$ -form such that $\Theta|_{\xi}\leq dv_{\xi}$ for any oriented $n$ -plane $\xi$ in the tangent bundle of $M$ , then $\Theta$ is called a calibration on $M$ , where $dv_{\xi}$ denotes the volume form on $\xi$ . An oriented $n$ -submanifold $L$ of $M$ is called calibrated by the calibration $\Theta$ , if $\Theta|_{L}$ equals to the volume form of $g|_{L}$ on $L$ . Mclean studied the deformation theory of calibrated submanifolds in [28].

Holomorphic submanifolds in Kähler manifolds, and special lagrangian submanifolds in Calabi-Yau manifolds are examples of calibrated submanifolds (c.f. [21]). If $(M,\omega,J,g)$ is a Kähler $n$ -manifold, then $\frac{1}{m!}\omega^{m}$ , $m\leq n$ , are calibrations on $M$ , and holomorphic $m$ -submanifolds are calibrated by $\frac{1}{m!}\omega^{m}$ . If $(M,\omega,J,g,\Omega)$ is a Ricci-flat Calabi-Yau $n$ -manifold, then, for any $\theta\in\mathbb{R}$ , ${\rm Re}e^{\sqrt{-1}\theta}\Omega$ is a calibration on $M$ , and a special lagrangian submanifold $L$ of phase $\theta$ is calibrated by ${\rm Re}e^{\sqrt{-1}\theta}\Omega$ .

In [16], a volume comparison theorem for calibrated submanifolds was obtained.

[05E7]

Theorem 6.1 (Theorem 2.0.1. in [16]).

Let $(M,g)$ be a closed Riemannian manifold, $\Theta$ be a calibration n-form, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{B_{g}(p,2\pi)}K_{g}\leq 1,

and there is a submanifold $L$ calibrated by $\Theta$ such that $p\in L$ . Then

Vol_{g}(B_{g}(p,r)\cap L)\geq Vol_{h_{1}}(B_{h_{1}}(r)),

for any $r\leq\min\{i_{g}(p),\pi\}$ , where $h_{1}$ denotes the standard metric on $S^{n}$ with constant curvature 1, and $B_{h_{1}}(r)$ denotes a metric $r$ -ball in $S^{n}$ .

By this theorem, we obtain the following estimate for injectivity radius:

[05E8]

Corollary 6.2.

Let $(M,g)$ be a closed Riemannian manifold, $\Theta$ be a calibration n-form, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{B_{g}(p,2\pi)}K_{g}\leq 1,

and there is a submanifold $L$ calibrated by $\Theta$ such that $\dim_{\mathbb{R}}L=n$ , $p\in L$ , and

\int_{L}\Theta<\frac{\pi}{2n}\varpi_{n-1},

where $\varpi_{n-1}$ is the volume of $S^{n-1}$ with the standard metric of constant curvature 1. Then the injectivity radius $i_{g}(p)$ of $(M,g)$ at $p$ satisfies that

i_{g}(p)^{n}\leq\frac{n\pi^{n-1}}{2^{n-1}\varpi_{n-1}}\int_{L}\Theta.

[05E9]

Proof.

By Theorem 6.1, we have

Vol_{h_{1}}(B_{h_{1}}(r))\leq Vol_{g}(B_{g}(p,r)\cap L)\leq Vol_{g}(L)=\int_{L}\Theta,

for any $r\leq\min\{i_{g}(p),\frac{\pi}{2}\}$ , where $h_{1}$ denotes the standard metric on $S^{n}$ with constant curvature 1, and $B_{h_{1}}(r)$ denotes a metric $r$ -ball in $S^{n}$ . Since $h_{1}=dr^{2}+\sin^{2}rh_{S^{n-1}}$ where $h_{S^{n-1}}$ is the standard metric on $S^{n-1}$ with constant curvature 1, we obtain $\sin r\geq\frac{2}{\pi}r$ , and

\frac{2^{n-1}}{n\pi^{n-1}}r^{n}\varpi_{n-1}\leq\int_{0}^{r}\sin^{n-1}rdr\varpi_{n-1}=Vol_{h_{1}}(B_{h_{1}}(r))\leq\int_{L}\Theta.

If $i_{g}(p)\geq\frac{\pi}{2}$ , by letting $r=\frac{\pi}{2}$ , we obtain

\frac{\pi}{2n}\varpi_{n-1}\leq\int_{L}\Theta<\frac{\pi}{2n}\varpi_{n-1},

which is a contradiction. Thus $i_{g}(p)<\frac{\pi}{2}$ . By letting $r=i_{g}(p)$ , we obtain

i_{g}(p)^{n}\leq\frac{n\pi^{n-1}}{2^{n-1}\varpi_{n-1}}\int_{L}\Theta.

∎

We obtain Theorem 1.5 by applying the above corollary to special lagrangian submanifolds in Ricci-flat Calabi-Yau manifolds. Another obvious application of Corollary 6.2 is to estimate injectivity radiuses by volumes of holomorphic submanifolds, which has independent interests.

[05EA]

Corollary 6.3.

Let $(M,\omega,J,g)$ be a closed Kähler n-manifold, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{M}K_{g}\leq 1,

and there is a smooth holomorphic m-submanifold $N$ such that $p\in N$ , and

\int_{N}\omega^{m}<\frac{(m-1)!\pi}{2}\varpi_{m-1}.

Then the injectivity radius $i_{g}(p)$ of $(M,g)$ at $p$ satisfies that

i_{g}(p)^{m}\leq\frac{\pi^{m-1}}{(m-1)!2^{m-1}\varpi_{m-1}}\int_{N}\omega^{m}.

By combining this corollary and the result in [6], there are $F$ -structures of positive rank on the regions of Kähler manifolds with bounded curvature and fibred by holomorphic submanifolds with small volumes.

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