#
SO(n)-Invariant special Lagrangian submanifolds of ℂ ^{n+1}
with fixed loci

Journal Article (Journal Article)

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. © The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2006. n n+1 n n+1 n+1

### Full Text

### Duke Authors

### Cited Authors

- Bryant, RL

### Published Date

- January 1, 2006

### Published In

### Volume / Issue

- 27 / 1

### Start / End Page

- 95 - 112

### Electronic International Standard Serial Number (EISSN)

- 1860-6261

### International Standard Serial Number (ISSN)

- 0252-9599

### Digital Object Identifier (DOI)

- 10.1007/s11401-005-0368-5

### Citation Source

- Scopus