Collapsing of Calabi-Yau manifolds and special lagrangian submanifolds

TL;DR

This paper explores the link between special lagrangian submanifolds and the collapsing behavior of Calabi-Yau manifolds, showing that their existence influences and is influenced by geometric collapse.

Contribution

It constructs special lagrangian fibrations on certain regions and demonstrates that small-volume lagrangian submanifolds imply collapsing regions in Calabi-Yau manifolds.

Findings

Construction of special lagrangian fibrations in collapsed regions

Small lagrangian submanifolds indicate collapsing regions

Bidirectional relationship between lagrangian submanifolds and collapsing

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.