Deformations of Special Lagrangian Submanifolds; An Approach via Fredholm Alternative

TL;DR

This paper revisits the deformation theory of special Lagrangian submanifolds in symplectic manifolds with non-integrable structures, demonstrating the surjectivity of the linearized operator via the Fredholm Alternative, offering a new proof method.

Contribution

It introduces an alternative proof for the deformation space smoothness using the Fredholm Alternative instead of the implicit function theorem.

Findings

Confirmed the surjectivity of the linearized operator using Fredholm Alternative.

Established the deformation space as a smooth manifold of dimension H^1(L).

Provided a new methodological approach for deformation analysis.

Abstract

In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H^1(L), the space of harmonic 1-forms on L. We proved this first by showing that the linearized operator for the deformation map is surjective and then applying the Banach space implicit function theorem. In this paper, we obtain the same surjectivity result by using a different method, the Fredholm Alternative, which is a powerful tool for compact operators in linear functional analysis.