Collapsing of Calabi-Yau manifolds and special Lagrangian submanifolds

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, conversely, 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.