Survey on real forms of the complex A2(2)-Toda equation and surface theory

Josef F Dorfmeister,Erxiao Wang,Shimpei Kobayashi,Walter Freyn

doi:10.1515/coma-2019-0011

Abstract

AbstractThe classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.

Highlights

Following the important work of Zakharov-Shabat [40] and Ablowitz-Kaup-Newell-Segur [1] in the 1970s, systematic constructions of hierarchies of integrable di erential equations were developed. They were associated to a complex simple Lie algebra with various reality conditions given by nite order automorphisms
Mikhailov [27] rst studied their reductions with various reality conditions given by nite order automorphisms
It is amazing that these two equations have already appeared in classical di erential geometry for constant negative Gauss curvature surfaces and constant mean curvature surfaces

Summary

Introduction

Following the important work of Zakharov-Shabat [40] and Ablowitz-Kaup-Newell-Segur [1] in the 1970s, systematic constructions of hierarchies of integrable di erential equations were developed. We discuss a loop group formulation of minimal Lagrangian surfaces in the complex projective plane CP. It is clear that the Maurer-Cartan form αCP = FC−P dFCP = UCP dz + VCP dz of the theory minimal Lagrangian surface is given by. From the discussion just above we derive a family of Maurer-Cartan forms in (1.14) of minimal Lagrangian surfaces from D to CP. They can be computed explicitly as λ dz, for. We discuss a loop group formulation of minimal Lagrangian surfaces in the complex hyperbolic plane CH.

The -form αCH satis es the Maurer-Cartan equations if and only if

We may conjugate the complex frame to a real SL R-frame:

The minus subgroup: