SO(n)-Invariant Special Lagrangian Submanifolds of ℂ ⁿ⁺¹ with Fixed Loci*

Abstract

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.

Keywords

Calibrations / Special Lagrangian submanifolds / 53C42 / 35A20