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

Abstract

This is the first of three papers math.DG/0111326, math.DG/0204343 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a U(1)-invariant SL 3-fold. Then |z_1|^2-|z_2|^2=2a on N for some real number a. 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. In the sequels we extend these results to the singular case a=0. The next paper math.DG/0111326 proves existence and uniqueness of continuous weak solutions to the two Dirichlet problems when a=0. This gives existence and uniqueness of a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The final paper math.DG/0204343 studies the nature of the singularities that arise, and constructs U(1)-invariant special Lagrangian fibrations of open sets in C^3.