Stability of harmonic maps between symmetric spaces

1982 ◽  
pp. 130-137 ◽  
Author(s):  
T. Nagano
Keyword(s):  
Symmetric Spaces ◽  
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.

scholarly journals Integrability and vesture for harmonic maps into symmetric spaces

2016 ◽  
Vol 28 (03) ◽  
pp. 1650006 ◽  
Author(s):  
Shabnam Beheshti ◽  
Shadi Tahvildar-Zadeh
Keyword(s):  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Dressing Method ◽  
Special Cases ◽  
Einstein Vacuum ◽  
Completely Reducible

After formulating the notion of integrability for axially symmetric harmonic maps from [Formula: see text] into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding [Formula: see text]-solitonic harmonic maps into a non-compact Grassmann manifold [Formula: see text] is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases [Formula: see text] and [Formula: see text] and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

scholarly journals ADDING A UNITON VIA THE DPW METHOD

2009 ◽  
Vol 20 (08) ◽  
pp. 997-1010 ◽  
Author(s):  
N. CORREIA ◽  
R. PACHECO
Keyword(s):  
Finite Type ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Point Of View ◽  
Type Property ◽  
Riemannian Symmetric Spaces

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

scholarly journals Equivariant harmonic maps depend real analytically on the representation

2021 ◽  
Author(s):  
Ivo Slegers
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Main Tool ◽  
Target Space ◽  
Fixed Target ◽  
Compact Type ◽  
Proper Action ◽  
Real Analytic

AbstractWe consider harmonic maps into symmetric spaces of non-compact type that are equivariant for representations that induce a free and proper action on the symmetric space. We show that under suitable non-degeneracy conditions such equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [4] can be applied.

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