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Publications [#243387] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, SO(n)Invariant special Lagrangian submanifolds of ℂ n+1 with fixed loci,
Chinese Annals of Mathematics, Series B, vol. 27 no. 1
(January, 2006),
pp. 95112, Springer Nature [MR2209954], [math.DG/0402201], [doi]
(last updated on 2019/05/27)
Abstract: 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 realanalytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular realanalytic 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 SpringerVerlag Berlin Heidelberg 2006.
Keywords: calibrations, special Lagrangian submanifolds


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