Special Lagrangian 3-folds and integrable systems

Abstract

Special Lagrangian 3-folds 191 related geometry is [22].We begin by defining calibrations and calibrated submanifolds, following Harvey and Lawson [13].Definition 2.1 Let (M,g) be a Riemannian manifold.An oriented tangent k-plane Von M is a vector subspace V of some tangent space TxM to M with dim V = k, equipped with an orientation.If V is an oriented tangent k-plane on M then glv is a Euclidean metric on V, so combining glv with the orientation on V gives a natural volume form volv on V, which is a k-form on V. Now let cp be a closed k-form on M. We say that cp is a calibration on M if for every oriented k-plane V on M we have 'Piv ~ volv.Here 'Piv = a• volv for some a E JR, and 'Piv ~ volv if a ~ 1.Let N be an oriented submanifold of M with dimension k.Then each tangent space TxN for x E N is an oriented tangent k-plane.We say that N isIt is easy to show that calibrated submanifolds are automatically minimal submanifolds [13, Th.II.4.2].Here is the definition of special Lagrangian submanifolds in em, taken from [13, §III].Definition 2.2 Let em have complex coordinates (zl' ... 'Zm), and define a metric g, a real 2-form w and a complex m-form non em byThen Re 0 and Im 0 are real m-forms on em.Let L be an oriented real submanifold of em of real dimension m, and let e E [0, 2n).We say that L is a special Lagrangian submanifold of em if L is calibrated with respect to Re 0, in the sense of Definition 2.1.We will often abbreviate 'special Lagrangian' by 'SL', and 'm-dimensional submanifold' by 'mfold', so that we shall talk about SL m-folds in em.As in [19] there is also a more general definition of special Lagrangian submanifolds involving a phase eie, but we will not use it in this paper.Harvey and Lawson [13, Cor.III.l.ll] give the following alternative characterization of special Lagrangian submanifolds.Proposition 2.3.Let L be a real m-dimensional submanifold of em.Then L admits an orientation making it into an SL submanifold of em if and only if wiL = 0 and ImOIL = 0.