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

Published

Journal Article

Let SO(n) act in the standard way on ℂn and extend this action in the usual way to ℂn+1 = ℂ ⊕ ℂ n . It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂn+1 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. 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.

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