Special Lagrangian conifolds. II: Gluing constructions in \(\mathbb C^m\) (Q2843971)

In previous works of the author [Commun. Anal. Geom. 21, No. 1, 105--170 (2013; Zbl 1284.46031); Proc. Lond. Math. Soc. (3) 107, No. 1, 198--224 (2013; Zbl 1275.53048)] the analytic and geometric foundations on a general theory of SL conifolds in \(\mathbb C^m\) is presented, that are now used in the present paper for defining a gluing construction that produces new examples of AC SLs and of SL conifolds in \(\mathbb{C}^m\) (in previous terminology). This could not be obtained using Joyce methods only holding for compact SLs, although many arguments are still inspired in Joyce's work, but using new tools as for example weighted Sobolev spaces adapted to neck regions and non-compact ends. This construction provides a parameterized family of AC SLs submanifolds by gluing a Lawlor neck into a neighbourhood of each intersection point of a finite number of SL planes satisfying Lawlor's angle conditions. This family can be seen as desingularization process defined by a parameter deformation of the initial family of transverse SL planes, where the parameters are given by the size of the necks, and shows how do transform any singular conifold into a smooth AC SL (in a weak sense) by replacing isolates CSs with AC ends, and how to attach new ends. This also provides the first examples of smooth SL conifolds with three planar ends at least and examples of SL conifolds with conic singularities that are not globally cones.