Surfaces in RN2−1 based on harmonic maps S2→CPN−1

Zakrzewski, W. J.

doi:10.1063/1.2815906

Cited by 8 publications

(9 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is worth mentioning that from the Veronese sequences we can obtain associated surfaces with constant Gaussian curvature as stated in [37]. However, the converse statement does not apply in general.…”

Section: Weierstrass Representationmentioning

confidence: 99%

SURFACES OBTAINED FROM ℂP^N-1 SIGMA MODELS

Grundland

Yurdusen

2008

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

In this paper, the Weierstrass technique for harmonic maps S 2 → CP N−1 is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP N−1 model equations are defined on the sphere S 2 and the associated action functional of this model is finite, then the generalized Weierstrass formula for immersion describes conformally parametrized surfaces in the su(N ) algebra. In particular, for any holomorphic or antiholomorphic solution of this model the associated surface can be expressed in terms of an orthogonal projector of rank (N − 1). The implementation of this method is presented for two-dimensional conformally parametrized surfaces immersed in the su(3) algebra. The usefulness of the proposed approach is illustrated with examples, including the dilation-invariant meron-type solutions and the Veronese solutions for the CP 2 model. Depending on the location of the critical points (zeros and poles) of the first fundamental form associated with the meron solution, it is shown that the associated surfaces are semi-infinite cylinders. It is also demonstrated that surfaces related to holomorphic and mixed Veronese solutions are immersed in R 8 and R 3 , respectively.

show abstract

Section: Weierstrass Representationmentioning

confidence: 99%

SURFACES OBTAINED FROM ℂP^N-1 SIGMA MODELS

Grundland

Yurdusen

2008

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

show abstract

“…It follows from the Bonnet theorem that the surfaces X k are determined uniquely up to Euclidean motions by their first fundamental forms (42) and their second fundamental forms…”

Section: Geometrical Aspects Of the Cp N−1 Modelmentioning

confidence: 99%

Invariant recurrence relations for {{{\mathbb C}}P^{N-1}} models

Goldstein¹,

Grundland²

2010

J. Phys. A: Math. Theor.

113

View full text Add to dashboard Cite

In this paper, we present invariant recurrence relations for the completely integrable CP N −1 Euclidean sigma model in two dimensions defined on the Riemann sphere S 2 when its action functional is finite. We determine the links between successive projection operators, wave functions of the linear spectral problem, and immersion functions of surfaces in the su(N ) algebra together with outlines of the proofs. Our formulation preserves the conformal and scaling invariance of these quantities. Certain geometrical aspects of these relations are described. We also discuss the singularities of meromorphic solutions of the CP N −1 model and show that they do not affect the invariant quantities. We illustrate the construction procedure through the examples of the CP 2 and CP 3 models.

show abstract

“…By Theorem 4.2 of [3], these complex functions satisfy ā3 a 4 − a 3 ā4 = 0. Differentiating (24) and applying property (P4), we see that f 2 • (f 0 × f −1 ) = 0, hence a 1 = 0. We also have…”

Section: The Parallel Surfacesmentioning

confidence: 98%

“…Eschenburg and Quast [11] replaced S 2 by an arbitrary Kähler symmetric space N = G/K of compact type and applied a natural generalization of Sym-Bobenko's formula [18] to the associated family of a harmonic map ϕ : Σ → N in order to obtain an immersion F of Σ in the Lie algebra g of G. This construction was subsequently generalized to primitive harmonic maps from Σ to generalized flag manifolds [21]. Surfaces associated to harmonic maps into complex projective spaces have also been constructed in [14,15,24].…”

Section: Introductionmentioning

confidence: 99%

Surfaces in $\mathbb{R}^7$ obtained from harmonic maps in $S^6$

Morais,

Pacheco

2016

Preprint

View full text Add to dashboard Cite

We will investigate the local geometry of the surfaces in the 7dimensional Euclidean space associated to harmonic maps from a Riemann surface Σ into S 6 . By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions ϕ : Σ → S 6 whose associated immersions F : Σ → R 7 belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.

show abstract

Surfaces in RN2−1 based on harmonic maps S2→CPN−1

Abstract: We show that many surfaces in RN2−1 can be generated by harmonic maps of S2→CPN−1. These surfaces are based on the projectors in CPN−1 which describe maps of S2→CPN−1. In the case when these maps form the Veronese sequence, all the surfaces have a constant curvature.

Cited by 8 publications

References 10 publications

SURFACES OBTAINED FROM ℂP^N-1 SIGMA MODELS

SURFACES OBTAINED FROM ℂP^N-1 SIGMA MODELS

Invariant recurrence relations for {{{\mathbb C}}P^{N-1}} models

Surfaces in $\mathbb{R}^7$ obtained from harmonic maps in $S^6$

Contact Info

Product

Resources

About

Surfaces in RN2−1 based on harmonic maps S2→CPN−1

Abstract: We show that many surfaces in RN2−1 can be generated by harmonic maps of S2→CPN−1. These surfaces are based on the projectors in CPN−1 which describe maps of S2→CPN−1. In the case when these maps form the Veronese sequence, all the surfaces have a constant curvature.

Cited by 8 publications

References 10 publications

SURFACES OBTAINED FROM ℂPN-1 SIGMA MODELS

SURFACES OBTAINED FROM ℂPN-1 SIGMA MODELS

Invariant recurrence relations for {{{\mathbb C}}P^{N-1}} models

Surfaces in $\mathbb{R}^7$ obtained from harmonic maps in $S^6$

Contact Info

Product

Resources

About

SURFACES OBTAINED FROM ℂP^N-1 SIGMA MODELS

SURFACES OBTAINED FROM ℂP^N-1 SIGMA MODELS