Special Lagrangian m-folds in C^m with symmetries

TL;DR

This paper explores special Lagrangian submanifolds in complex Euclidean spaces with symmetries, providing explicit constructions of invariant cones that serve as models for singularities relevant to Mirror Symmetry and the SYZ conjecture.

Contribution

It introduces a method to construct G-invariant special Lagrangian cones in C^m using ODEs, expanding the known examples and aiding the understanding of singularities in Calabi-Yau manifolds.

Findings

Constructed a large family of G-invariant special Lagrangian cones

Solved the special Lagrangian equation as an ODE in G-orbits

Provided local models for singularities in Calabi-Yau manifolds

Abstract

This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models will be needed to understand Mirror Symmetry and the SYZ conjecture.