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.