SO(n)-Invariant Special Lagrangian Submanifolds of ℂ n+1 with Fixed Loci*
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Let SO(n) act in the standard way on ℂ n and extend this action in the usual way to ℂ n+1 = ℂ ⊕ ℂ n .
It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂ n+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.
KeywordsCalibrations Special Lagrangian submanifolds
MR (2000) Subject Classification53C42 35A20
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