Evolution Equations for Special Lagrangian 3-Folds in C3

Abstract

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m − 1)-submanifold P in Rⁿ, 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 affine maps φ _t : Rⁿ→ 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 C³. We find a one-to-one 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 C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuGqLXgBG0evaeXatLxBI9gBam% XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB% Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf% euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9% q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba% WaaqaafaaakeaaruWrPDwAaGGbciab-nfatbaa!3D86! ¹× R². 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 ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.