Minimal Lagrangian surfaces in $\mathbb{C}P^2$ via the loop group method Part II: The general case

TL;DR

This paper develops a loop group method to construct and analyze minimal Lagrangian surfaces in complex projective space, extending previous techniques to arbitrary Riemann surfaces and introducing new classes of such surfaces.

Contribution

It generalizes the construction of minimal Lagrangian immersions to all Riemann surfaces and introduces perturbed equivariant surfaces and their approximation by Delaunay cylinders.

Findings

Constructed minimal Lagrangian immersions from arbitrary Riemann surfaces.

Introduced perturbed equivariant minimal Lagrangian surfaces.

Showed approximation of these surfaces by Delaunay cylinders.

Abstract

We extend the techniques introduced in \cite{DoMaB1} for contractible Riemann surfaces to construct minimal Lagrangian immersions from arbitrary Riemann surfaces into $\mathbb{C}P^2$ via the loop group method. Based on the potentials of translationally equivariant minimal Lagrangian surfaces, we introduce perturbed equivariant minimal Lagrangian surfaces in $\mathbb{C}P^2$ and construct a class of minimal Lagrangian cylinders. Furthermore, we show that these minimal Lagrangian cylinders approximate Delaunay cylinders with respect to some weighted Wiener norm of the twisted loop group $\Lambda SU(3)_{\sigma}$.