Let SO（n） act in the standard way on C^n and extend this action in the usual way to C^n＋l = C＋ C^n. It is shown that a nonsingular special Lagrangian submanifold L→∪ C^n＋l that is invariant under this SO（n）-...Let SO（n） act in the standard way on C^n and extend this action in the usual way to C^n＋l = C＋ C^n. It is shown that a nonsingular special Lagrangian submanifold L→∪ C^n＋l that is invariant under this SO（n）-action intersects the fixed C→∪ C^n＋1 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. nonsingular SO（n）-invariant special Lagrangian nonsingular SO（n）-invariant special Lagrangian extensions in some open neighborhood of A. If A is connected, there exist n distinct extensions of A such that any embedded extension of A agrees with one of these n 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.展开更多
基金Project supported by Duke University via a research grant, the NSF via DMS-0103884the Mathematical Sciences Research Institute, and Columbia University.
文摘Let SO（n） act in the standard way on C^n and extend this action in the usual way to C^n＋l = C＋ C^n. It is shown that a nonsingular special Lagrangian submanifold L→∪ C^n＋l that is invariant under this SO（n）-action intersects the fixed C→∪ C^n＋1 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. nonsingular SO（n）-invariant special Lagrangian nonsingular SO（n）-invariant special Lagrangian extensions in some open neighborhood of A. If A is connected, there exist n distinct extensions of A such that any embedded extension of A agrees with one of these n 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.
