Ruled Special Lagrangian 3‐Folds In C 3

This is the fifth in a series of papers constructing explicit examples of special Lagrangian submanifolds in C m . A submanifold of C m is ruled if it is fibred by a family of real straight lines in C m . This paper studies ruled special Lagrangian 3‐folds in C 3 , giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko, and Bryant. Special Lagrangian cones in C 3 are automatically ruled, and each ruled special Lagrangian 3‐fold is asymptotic to a unique special Lagrangian cone. We study the family of ruled special Lagrangian 3‐folds N asymptotic to a fixed special Lagrangian 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 σ of N0 with the unit sphere S 5 in C 3 is a Riemann surface, and construct a ruled special Lagrangian 3‐fold N asymptotic to N0 for each holomorphic vector field w on σ. As corollaries of this we write down two large families of explicit special Lagrangian 3‐folds in C 3 depending on a holomorphic function on C, which include many new examples of singularities of special Lagrangian 3‐folds. We also show that each special Lagrangian T2‐cone N0 can be extended to a 2‐parameter family of ruled special Lagrangian 3‐folds asymptotic to N0, and diffeomorphic to T 2 ×R. 2000 Mathematical Subject Classification : 53C38, 53D12.