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

Abstract

This is the first of three papers studying special Lagrangian 3-submanifolds ( SL 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. Let N be such a U(1)-invariant SL 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. 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 C 3 , 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 C 3 .