On the Lifts of Minimal Lagrangian Submanifolds

TL;DR

This paper explores how minimal Lagrangian submanifolds can be lifted to calibrated submanifolds within various special geometric structures, revealing new insights into their geometric and topological properties.

Contribution

It introduces a framework for lifting minimal Lagrangian submanifolds to calibrated submanifolds in Calabi-Yau, G2, and Spin(7) structures, extending previous understanding.

Findings

Canonical line bundle supports integrable special structures

Lifts of minimal Lagrangian submanifolds are calibrated in these structures

Super-minimal surfaces admit special calibrated lifts

Abstract

We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this setting and hence can be considered as the special Lagrangian lift of $M$. For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by Bryant and Salamon, minimal surfaces with vanishing complex quartic form (super-minimal) admit lifts which are calibrated, i.e., associative, coassociative or Cayley respectively. The lifts in this case can be considered as the tangential lifts or normal lifts of the minimal surface adapted to the quaternionic bundle structure.