Evolution equations for special Lagrangian 3-folds in C^3

Dominic Joyce

Published 2000-10-03, updated 2001-08-01Version 2

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper in the series, math.DG/0008155, defined the idea of evolution data, which includes an (m-1)-submanifold P in R^n, and constructed a family of special Lagrangian m-folds N in C^m, which are swept out by the image of P under a 1-parameter family of linear or affine maps phi_t : R^n -> C^m, satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C^3. We find a 1-1 correspondence between sets of evolution data with m=3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C^3. Our main results are a number of new families of special Lagrangian 3-folds in C^3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R^3 or S^1 x R^2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind. We hope these 3-folds will be helpful in understanding singularities of compact special Lagrangian 3-folds in Calabi-Yau 3-folds. This will be important in resolving the SYZ conjecture in Mirror Symmetry.