SO(n)-Invariant special Lagrangian submanifolds of ℂ n+1 with fixed loci
Repository Usage Stats
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.
Published Version (Please cite this version)10.1007/s11401-005-0368-5
Publication InfoBryant, RL (2006). SO(n)-Invariant special Lagrangian submanifolds of ℂ n+1 with fixed loci. Chinese Annals of Mathematics. Series B, 27(1). pp. 95-112. 10.1007/s11401-005-0368-5. Retrieved from https://hdl.handle.net/10161/12682.
This is constructed from limited available data and may be imprecise. To cite this article, please review & use the official citation provided by the journal.
More InfoShow full item record
Phillip Griffiths Professor of Mathematics
My research concerns problems in the geometric theory of partial differential equations. More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy. Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems. Because of their built-in invariance properties, these systems often have specia