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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 [MR2209954], [math.DG/0402201]
(last updated on 2018/10/14)
Abstract: Let SO(n) act in the standard way on C^n and extend this action in the usual
way to C^{n+1}. It is shown that nonsingular special Lagrangian submanifold
L in C^{n+1} that is invariant under this SO(n)action intersects the fixed line
C in a nonsingular realanalytic arc A (that may be empty). If n>2, then A has
no compact component. Conversely, an embedded, noncompact nonsingular
realanalytic arc A in 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 Gerard and Tahara to prove the existence of the extension.
Keywords: calibrations, special Lagrangian submanifolds


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