Formal Killing fields for minimal Lagrangian surfaces in complex space forms

Abstract

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the differential algebraic inductive formulas for a pair of loop algebra $\mathfrak{sl}(3,\mathbb{C})[[\lambda]]$-valued canonical formal Killing fields. As a result, we give a complete classification of the (infinite sequence of) Jacobi fields for the minimal Lagrangian system. We also obtain an infinite sequence of higher-order conservation laws from the components of the formal Killing fields.