SO(n)-invariant special Lagrangian submanifolds of C^{n+1} with fixed loci
Abstract
Description
Let SO(n) act in the standard way on C^n and extend this action in the usual way to C^{n+1}.
It is shown that a nonsingular special Lagrangian submanifold L in C^{n+1} that is invariant under this SO(n)-action intersects the fixed line C in a nonsingular real-analytic arc A (that may be empty). If n>2, then A has no compact component.
Conversely, an embedded, noncompact nonsingular real-analytic arc A in 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 Gerard and Tahara to prove the existence of the extension.
19 pages, no figures. Miinor update: some misprints fixed and a reference added
19 pages, no figures. Miinor update: some misprints fixed and a reference added