Second Order Families of Special Lagrangian Submanifolds in C^4

TL;DR

This paper classifies special Lagrangian 4-folds in C^4 with nontrivial symmetry groups, extending previous 3-dimensional results, using exterior differential systems and geometric analysis to explicitly describe these submanifolds.

Contribution

It extends the classification of special Lagrangian submanifolds to four dimensions with symmetry, providing explicit descriptions using Cartan-Kahler theory.

Findings

Classified special Lagrangian 4-folds with nontrivial SO(4) stabilizers.

Used exterior differential systems to prove existence of such families.

Provided explicit geometric descriptions of many cases.

Abstract

This paper extends to dimension 4 the results in the article "Second Order Families of Special Lagrangian 3-folds" by Robert Bryant. We consider the problem of classifying the special Lagrangian 4-folds in C^4 whose fundamental cubic at each point has a nontrivial stabilizer in SO(4). Points on special Lagrangian 4-folds where the SO(4)-stabilizer is nontrivial are the analogs of the umbilical points in the classical theory of surfaces. In proving existence for the families of special Lagrangian 4-folds, we used the method of exterior differential systems in Cartan-Kahler theory. This method is guaranteed to tell us whether there are any families of special Lagrangian submanifolds with a certain symmetry, but does not give us an explicit description of the submanifolds. To derive an explicit description, we looked at foliations by submanifolds and at other geometric particularities. In this manner, we settled many of the cases and described the families of special Lagrangian submanifolds in an explicit way.