Special Lagrangian m-folds in ℂm with symmetries

By D. Joyce. 2000

115 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/04df955ca41b222cf212fb7c216a4f36d6f441e7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Lectures on special Lagrangian geometry (2001, methodology): “We now describe five methods of constructing special Lagrangian m-folds in C, drawn from papers by the author [15, 16, 18, 19, 20, 21, 22, 23], Bryant [1], Castro and Urbano [2], Goldstein [4, 5], Harvey [10, p.”
M theory and singularities of exceptional holonomy manifolds (2004, unknown): “If we choose to parametrize Re(C3) = R3 and Im(C3) = R3 by 3-vectors ~ φ1 and ~ φ2, respectively, then one can explicitly describe L as the zero locus of the polynomial relations [71]:”
Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in $$\mathbb{C}^n$$ (2006, methodology): “The equation Im (γ n) = Const and its relations with special Lagrangian submanifolds was also considered independently in Haskins [15] and Joyce [16].”