U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions

Abstract

This is the last of three papers studying special Lagrangian 3- submanifolds ( SLV 3- folds) N in C 3 invariant under the U(1)-action e iθ :( z 1, z 2, z 3)↦(e iθ z 1,e iθ z 2, z 3), using analytic methods. If N is such a 3-fold then | z 1| 2−| z 2| 2=2 a on N for some a∈ R . Locally, N can be written as a kind of graph of functions u,v: R 2→ R satisfying a nonlinear Cauchy–Riemann equation depending on a, so that u+ iv is like a holomorphic function of x+ iy. The first paper studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy–Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C 3 , with boundary conditions. The second paper extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in C 3 , with boundary conditions. This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicity n>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in C 3 , including singular fibres.