Deformations of Asymptotically Conical Special Lagrangian Submanifolds

The naive approach is to parametrize these deformations as the zero-set of a “mean curvature operator”, then study them using the implicit function theorem. However, this entails a good understanding of the Jacobi operator of the initial submanifold Σ, which in general is not possible. The work of Oh and, more recently, of McLean (cfr. [Oh], [ML]) shows that, in the “right” geometric context, the problem simplifies and sometimes becomes tractable. We are thus lead to the study of minimal Lagrangian submanifolds in Kaehler-Einstein (KE) ambient spaces, and of special Lagrangian (SL) submanifolds in Calabi-Yau (CY) manifolds. In section 2 of this article, we provide all the necessary definitions and some basic examples of non-compact, “asymptotically conical” SL and CY manifolds. In [ML], McLean shows that every “infinitesimal SL deformation” of a compact SL submanifold is “integrable”; ie, it generates actual SL deformations. A corollary of this is, in the compact case, that the set