scholarly journals Minimal Lagrangian surfaces in $\Bbb S\sp 2 x $\Bbb S\sp 2$

2007 ◽  
Vol 15 (2) ◽  
pp. 217-248 ◽  
Author(s):  
Ildefonso Castro ◽  
Francisco Urbano
Keyword(s):  
Lagrangian Surfaces ◽  
Minimal Lagrangian Surfaces

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

2019 ◽  
Vol 6 (1) ◽  
pp. 194-227 ◽  
Author(s):  
Josef F. Dorfmeister ◽  
Walter Freyn ◽  
Shimpei Kobayashi ◽  
Erxiao Wang
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Loop Group ◽  
Additional Parameter ◽  
Lagrangian Surfaces ◽  
Affine Spheres ◽  
Reductive Lie Groups ◽  
Real Forms ◽  
Minimal Lagrangian Surfaces

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.

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