A moduli space of minimal affine Lagrangian submanifolds

TL;DR

This paper proves that the space of all connected, compact, orientable, embedded minimal affine Lagrangian submanifolds in a complex equiaffine space forms an infinite dimensional Frechet manifold, extending to Lagrangian surfaces in almost Kaehler 4-manifolds.

Contribution

It establishes the infinite dimensional Frechet manifold structure of the moduli space of minimal affine Lagrangian submanifolds and Lagrangian surfaces in specific geometric contexts.

Findings

Moduli space of minimal affine Lagrangian submanifolds is an infinite dimensional Frechet manifold.

Moduli space of Lagrangian surfaces in almost Kaehler 4-manifolds is an infinite dimensional Frechet manifold.

Results hold for connected, compact, orientable, embedded submanifolds.

Abstract

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces in an almost Kaehler 4-dimensional manifold forms an infinite dimensional Frechet manifold (if it is not the empty set).