Special Lagrangian m-folds in C^m with symmetries

2000/08/31 by Dominic Joyce · 2 citations
Mathematics · #math.DG

paper · pdf

published as Duke Mathematical Journal 115 (2002), 1-51. · 44 pages, LaTeX; (v4) minor corrections and improvements

arxiv created 2001/07/31 · arxiv updated 2009/11/30

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.

Cited by

• Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two
• Lagrangian Translating Solitons and Special Lagrangians in ℂ^m with Symmetries

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