scholarly journals Weierstrass type representation of harmonic maps into symmetric spaces

1998 ◽  
Vol 6 (4) ◽  
pp. 633-668 ◽  
Author(s):  
J. Dorfmeister ◽  
F. Pedit ◽  
H. Wu
Keyword(s):  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Type Representation ◽  
Weierstrass Type Representation

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.

Classes of generalized Weingarten surfaces in the Euclidean 3-space

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Diogo G. Dias ◽  
Armando M. V. Corro
Keyword(s):  
Fixed Point ◽  
Distance Function ◽  
Special Class ◽  
Support Function ◽  
The Other ◽  
Isolated Singularity ◽  
Type Representation ◽  
Weingarten Surfaces ◽  
Parallel Surfaces ◽  
Weierstrass Type Representation

AbstractWe present surfaces with prescribed normal Gaussmap. These surfaces are obtained as the envelope of a sphere congruencewhere the other envelope is contained in a plane. We introduce classes of surfaces that generalize linear Weingarten surfaces, where the coefficients are functions that depend on the support function and the distance function from a fixed point (in short, DSGW-surfaces). The linear Weingarten surfaces, Appell’s surfaces and Tzitzeica’s surfaces are all DSGW-surfaces. From them we obtain new classes of DSGWsurfaces applying inversions, dilatations and parallel surfaces. For a special class of DSGW-surfaces, which is invariant under dilatations and inversions, we obtain a Weierstrass type representation (in short, EDSG Wsurfaces). As applications we classify the EDSGW-surfaces of rotation and present a 2-parameter family of complete cyclic EDSGW-surfaces with an isolated singularity and foliated by non-parallel planes.

scholarly journals Pseudospherical surfaces of low differentiability

2016 ◽  
Vol 16 (1) ◽  
pp. 1-20
Author(s):  
Josef F. Dorfmeister ◽  
Ivan Sterling
Keyword(s):  
Gauss Curvature ◽  
Type Representation ◽  
Pseudospherical Surfaces ◽  
Weierstrass Type Representation

AbstractWe continue our investigations into Toda’s algorithm [14; 3], which gives a Weierstrass-type representation of Gauss curvature K = −1 surfaces in R

A class of generalized special Weingarten surfaces

2019 ◽  
Vol 30 (14) ◽  
pp. 1950075
Author(s):  
Armando M. V. Corro ◽  
Diogo G. Dias ◽  
Carlos M. C. Riveros
Keyword(s):  
Fixed Point ◽  
Holomorphic Functions ◽  
Gauss Map ◽  
Type Representation ◽  
The Third ◽  
Weingarten Surfaces ◽  
Lines Of Curvature ◽  
Third Fundamental Form ◽  
Pass Through ◽  
Weierstrass Type Representation

In [Classes of generalized Weingarten surfaces in the Euclidean 3-space, Adv. Geom. 16(1) (2016) 45–55], the authors study a class of generalized special Weingarten surfaces, where coefficients are functions that depend on the support function and the distance function from a fixed point (in short EDSGW-surfaces), this class of surfaces has the geometric property that all the middle spheres pass through a fixed point. In this paper, we present a Weierstrass type representation for EDSGW-surfaces with prescribed Gauss map which depends on two holomorphic functions. Also, we classify isothermic EDSGW-surfaces with respect to the third fundamental form parametrized by planar lines of curvature. Moreover, we give explicit examples of EDSGW-surfaces and isothermic EDSGW-surfaces.

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.

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