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

AbstractThis is the first of three papers studying special Lagrangian 3-submanifolds (SL 3-folds) N in C3 invariant under the U(1)-action eiθ:(z1,z2,z3)↦(eiθz1,e−iθz2,z3), using analytic methods.Let N be such a U(1)-invariant SL 3-fold. Then |z1|2−|z2|2=2a on N for some a∈R. Locally, N can be written as a kind of graph of functions u,v:R2→R satisfying a nonlinear Cauchy–Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N.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