Special Lagrangian m-folds in ℂm with symmetries

Abstract

This 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 we 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 ordinary differential equation (ODE) in G-orbits and solving the ODE, we find a large family of distinct, G-invariant special Lagrangian cones on T^m−2 in ℂ^m. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models are needed to understand mirror symmetry and the Strominger-Yau-Zaslow (SYZ) conjecture.