SO(n)-INVARIANT SPECIAL LAGRANGIAN SUBMANIFOLDS OF C WITH FIXED LOCI

Let SO(n) act in the standard way on C and extend this action in the usual way to C = C ⊕ C. It is shown that a nonsingular special Lagrangian submanifold L ⊂ C that is invariant under this SO(n)-action intersects the fixed C ⊂ C in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ C lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear pde and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension.