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

(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, $\dim_{\mathbb{R}}L=n$,

(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.

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].

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

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

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.