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By Seares F.H., van Rhijn P.J.
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Case (a) above, with A = 0, corresponds to |C| = |D|, and in this case N is a subset of an affine special Lagrangian 3-plane R3 in C3 . If |C| = |D| then N is an embedded submanifold diffeomorphic to R3 , with coordinates (x1 , x2 , t). The two cases C = 0 and D = 0 are constructed by [7, Prop. 3] with n = 2, m = 3 and G = R, as in [7, Ex. 6] and case (c) above, with the symmetry group G of N acting by (x1 , x2 , t) → (x1 , x2 , t + c). Constructing special Lagrangian m-folds in Cm 797 Acknowledgements.
R. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47–157 6. D. Joyce, On counting special Lagrangian homology 3-spheres, hep-th/9907013, 1999 7. D. DG/0008021, 2000 8. D. DG/0010036, 2000 9. D. DG/0011179, 2000 10. D. DG/0012060, 2000 11. D. DG/0101249, 2001 12. G. Lawlor, The angle criterion, Inventiones math. 95 (1989), 437–446 13. A. -T. Yau, E. Zaslow, Mirror symmetry is T-duality, Nucl. Phys. B479 (1996), 243–259.
Let us parametrize the circle C+ with a parameter s, so that C+ = x1 (s), x2 (s), x3 (s) : s ∈ R . This gives a parametrization Φ : R2 → Σ+ of Σ+ , by Φ : (s, t) → w1 (t)x1 (s), w2 (t)x2 (s), w3 (t)x3 (s) . 4) We shall calculate the conditions upon xj (s) for Φ to be conformal, and solve them. Since the xj (s) satisfy α1 x12 + α2 x22 + α3 x32 = 1 and x12 − x22 − x32 = 0, differentiating with respect to s gives α1 x1 x˙1 + α2 x2 x˙2 + α3 x3 x˙3 = 0 and x1 x˙1 − x2 x˙2 − x3 x˙3 = 0, where ‘ ˙ ’is dsd .