Ruled special Lagrangian 3-folds in C³

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This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in Cᵐ. A submanifold of Cᵐ is ruled if it is fibred by a family of real straight lines in Cᵐ. This paper studies ruled special Lagrangian 3-folds in C³, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko and Bryant. An important class of ruled SL 3-folds is the special Lagrangian cones in C³. Each ruled SL 3-fold is asymptotic to a unique SL cone. We study the family of ruled SL 3-folds N asymptotic to a fixed SL cone N₀. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection Sigma of N₀ with the unit sphere in C³ is a Riemann surface, and construct a ruled SL 3-fold N asymptotic to N₀ for each holomorphic vector field w on Sigma. As corollaries of this we write down two large families of explicit SL 3-folds depending on a holomorphic function on C, which include many new examples of singularities of SL 3-folds. We also show that each SL T² cone N₀ can be extended to a 2-parameter family of ruled SL 3-folds asymptotic to N₀, and diffeomorphic to T² x R.