Calibrated Embeddings in the Special Lagrangian and Coassociative Cases
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G2-manifold, even as the fixed locus of an anti-G2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated sub-manifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.
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