Special Lagrangian conifolds, I: moduli spaces

Abstract

We discuss the deformation theory of special Lagrangian (SL) conifolds in ℂm. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi‐Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non‐compact, asymptotically conical, ends. Our main theorem is the natural next step in the chain of results initiated by McLean [Comm. Anal. Geom. 6 (1998) 705–747] and continued by Pacini [Pacific J. Math. 215 (2004) 151–181] and Joyce [Ann. Global Anal. Geom. 25 (2004) 301–352]. We emphasize a unifying framework for studying the various cases and discuss analogies and differences between them. This paper also lays down the geometric foundations for our paper Pacini [Proc. LMS to appear] concerning gluing constructions for SL conifolds in ℂm.

Special Lagrangian conifolds, II: gluing constructions in ℂ m

“…We now need to review the deformation theory of Lagrangian conifolds, following [14,24]. It is useful to do this in several steps.…”

“…The first part of the claim is a direct consequence of the definitions so we only need to check equation (4.3). We will use the same methods already used in Section 2 and in [26,Subsection 4.2] except that, instead of concentrating on the behaviour as r → 0, we concentrate on what happens in the subset Σ i × [t τ i , 2t τ i ]. Using equations (4.2) and (4.1), we see that,…”

“…It is also Part II of a multi-step project aiming to set up a general theory of SL conifolds. Two other papers related to this project are currently available [25,26] (see also [24]). The first of these papers provides the analytic foundations for our gluing construction, the second provides the geometric foundations, but actually each paper is self-contained and has its own, independent, focus.…”

“…Our theorems set only two restrictions on the SL conifolds themselves, as follows. Theorem 6.10 requires the remaining singularities to be 'stable': this assumption is rather natural and has already appeared in a previous work of Joyce and Haskins; see also [23,26]. The second restriction is as follows.…”

“…This is particularly true when dealing with the more geometrical aspects of these gluing theorems. In particular, the Lagrangian neighbourhoods we use in Subsection 2.3 to set up the gluing problem go mostly back to Joyce, with minor adaptations and changes introduced in [24]. The quadratic estimates of Section 5 also originate in the work of Joyce, although a certain amount of extra work is needed to adapt them to the presence of CS and AC ends.…”