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

Abstract

The 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 A (2) 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 CP 2 , minimal Lagrangian surfaces in CH 2 , timelike minimal Lagrangian surfaces in CH 2 1 , proper definite affine spheres in R 3 and proper indefinite affine spheres in R 3 , respectively.

Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

“…The present paper deals with the Lie group SL 3 C with an outer automorphismσ and some anti-linear involutionτ . It is known [16] that up to isomorphisms the Lie algebra and the order 6 automorphism are uniquely determined, see [8,Section 7] for details. Therefore, in our discussion above we could only change the anti-linear involution τ on Λsl 3 C σ , the so-called the real form involution.…”

“…(c) Actually, when trying to cover all surface classes falling under the scheme outlined above one also needs to consider what happens if one considers an anti-linear automorphism which is conjugated by an inner automorphism such that the induced anti-linear automorphism of the loop group/loop algebra still commutes withσ, see [8,Section 7]. As a matter of fact, such cases did already occur in the paper [18] and will also occur at least in case (2) above.…”

“…(2) 2 by Heintze-Groß [13] or Rousseau et al [3,4] it became clear, that the surface types mentioned above correspond exactly to four of the five types of inequivalent real forms, see [8]. For the case of the complex affine Kac-Moody algebra of type A…”

Ruh–Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $$\mathbb {C}P^2$$

“…

In this paper we investigate surfaces in CP 2 without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of an immersion from a surface to CP 2 into S 5 in Appendix A.(2) 2 type [5]. While σ arose naturally in classical geometric investigations, the question arose, whether also σ 2 and σ 3 have a simple geometric meaning.The starting point for an approach to this question was the paper [13], which investigated arbitrary immersions from Riemann surfaces to CP 2 without complex points.

…”

“…(2) 2 type [5]. While σ arose naturally in classical geometric investigations, the question arose, whether also σ 2 and σ 3 have a simple geometric meaning.…”