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

This is the last of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions 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 , 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 , 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 multiplicityn>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 , including singular fibres.