U(1)-invariant special Lagrangian 3-folds. I. Nonsingular solutions

This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C3, with two kinds of boundary conditions. The sequels extend these to the case a=0, study the singularities of the SL 3-folds that arise, and construct special Lagrangian fibrations of open sets in C3.