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Special Lagrangian m-folds in C^m with symmetries

Dominic Joyce
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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.

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Dominic Joyce. 2001-07-31. Special Lagrangian m-folds in C^m with symmetries. https://arxiv.org/abs/math/0008021

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