Special Lagrangian conifolds, II: gluing constructions in ℂ m

Abstract

We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the first (non-trivial) examples of SL conifolds which have a conical singularity but are not, globally, cones. We also obtain: (i) a desingularization procedure for transverse intersection and self-intersection points, using "Lawlor necks"; (ii) a construction which completely desingularizes any SL conifold by replacing isolated conical singularities with non-compact asymptotically conical (AC) ends; (iii) a proof that there is no upper bound on the number of AC ends of a SL conifold; (iv) the possibility of replacing a given collection of conical singularities with a completely different collection of conical singularities and of AC ends. As a corollary of (i) we improve a result by Arezzo and Pacard concerning minimal desingularizations of certain configurations of SL planes in C^m, intersecting transversally.