Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sⁿ by looking at the conormal bundle of appropriate submanifolds of Sⁿ. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in Rⁿ in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G₂ and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy. Mathematics Subject Classification (2000): 53-XX, 58-XX.     

Desingularization of Coassociative 4-Folds with Conical Singularities

Suppose that N is a compact coassociative 4-fold with a conical singularity in a 7-manifold M, with a G₂ structure given by a closed 3-form. We construct a smooth family, {N′(t) : t ∈ (0,τ)} for some τ > 0, of compact, nonsingular, coassociative 4-folds in M which converge to N in the sense of currents, in geometric measure theory, as t → 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4-fold in , dilated by t, then deforming the resulting compact 4-dimensional submanifold of M to the required coassociative 4-fold. Received: March 2007 Revision: June 2007 Accepted: June 2007     

Calibrated Embeddings in the Special Lagrangian and Coassociative Cases

Every closed, oriented, real analytic Riemannian3–manifold can be isometrically embedded as a specialLagrangian submanifold of a Calabi–Yau 3–fold, even as thereal locus of an antiholomorphic, isometric involution. Every closed,oriented, real analytic Riemannian 4–manifold whose bundle of self-dual2–forms is trivial can be isometrically embedded as a coassociativesubmanifold in a G₂-manifold, even as the fixed locus of ananti-G₂ involution.These results, when coupledwith McLean's analysis of the moduli spaces of such calibratedsubmanifolds, yield a plentiful supply of examples of compact calibratedsubmanifolds with nontrivial deformation spaces.     

G 2-manifolds and coassociative torus fibration

Let (φ ₀, g ₀) be a flat G ₂-structure on the torus T ⁷. For a certain finite group Γ-action on T ⁷ preserving the G ₂-structure, Joyce constructed a closed G ₂-manifold M from the resolution of the orbifold T ⁷/Γ. The main purpose of this paper is to prove that there exist global coassociative fibrations on open submanifolds of certain Joyce manifolds.