Harmonic maps into Lie groups and the multivalued Novikov functional

1993 ◽  
pp. 331-342
Author(s):  
A. V. Tyrin
Keyword(s):  
Lie Groups ◽  
Harmonic Maps

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.

GEOMETRY OF THE MODULI SPACES OF HARMONIC MAPS INTO LIE GROUPS VIA GAUGE THEORY OVER RIEMANN SURFACES

2001 ◽  
Vol 12 (03) ◽  
pp. 339-371
Author(s):  
MARIKO MUKAI-HIDANO ◽  
YOSHIHIRO OHNITA
Keyword(s):  
Critical Point ◽  
Lie Groups ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Energy Function ◽  
Riemann Surfaces ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Point Subset ◽  
Presymplectic Structure

This paper aims to investigate the geometry of the moduli spaces of harmonic maps of compact Riemann surfaces into compact Lie groups or compact symmetric spaces. The approach here is to study the gauge theoretic equations for such harmonic maps and the moduli space of their solutions. We discuss the S1-action, the hyper-presymplectic structure, the energy function, the Hitchin map, the flag transforms on the moduli space, several kinds of subspaces in the moduli space, and the relationship among them, especially the structure of the critical point subset for the energy function on the moduli space. As results, we show that every uniton solution is a critical point of the energy function on the moduli space, and moreover we give a characterization of the fixed point subset fixed by S1-action in terms of a flag transform.

On the geometrical properties of hypercomplex four-dimensional Lie groups

2020 ◽  
Vol 27 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Mehri Nasehi ◽  
Mansour Aghasi
Keyword(s):  
Lie Groups ◽  
Vector Fields ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Invariant Vector ◽  
Geometrical Properties ◽  
Invariant Vector Field ◽  
Left Invariant Vector ◽  
Totally Geodesic Hypersurfaces ◽  
Yamabe Solitons

AbstractIn this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.

scholarly journals A loop group method for affine harmonic maps into Lie groups

2016 ◽  
Vol 298 ◽  
pp. 207-253
Author(s):  
Josef F. Dorfmeister ◽  
Jun-ichi Inoguchi ◽  
Shimpei Kobayashi
Keyword(s):  
Lie Groups ◽  
Harmonic Maps ◽  
Loop Group ◽  
Group Method
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