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Publications [#243387] of Robert Bryant

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Papers Published

  1. 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. 95-112 [MR2209954], [math.DG/0402201]
    (last updated on 2017/12/18)

    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 real-analytic arc A (that may be empty). If n>2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic 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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