Harmonic maps of finite uniton number into G 2

Correia, N.; Pacheco, Rui

doi:10.1007/s00209-011-0849-z

Cited by 7 publications

(17 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3. Recently there have been several publications concerning harmonic maps into compact Lie groups and compact symmetric spaces which have used methods which are different from ours, see [18], [9], [25] and reference therein. Most of their work basically follows the techniques developed by Uhlenbeck [26] and Segal [24].…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

Wang¹

2017

Tohoku Math. J. (2)

View full text Add to dashboard Cite

The family of Willmore immersions from a Riemann surface into S n+2 can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in R n+2 and those which are not conformally equivalent to a minimal surface in R n+2 . On the level of their conformal Gauss maps into Gr 1,3 (R 1,n+3 ) = SO + (1, n + 3)/SO + (1, 3) × SO(n) these two classes of Willmore immersions into S n+2 correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of R 1,n+3 , contains a fixed lightlike vector or where it does not contain such a "constant lightlike vector". Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into S n+2 which are not conformal to a minimal surface in R n+2 . It turns out that our proof also works analogously for minimal immersions into the other space forms.

show abstract

Section: Proof Of Theorem 14mentioning

confidence: 99%

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

Wang¹

2017

Tohoku Math. J. (2)

View full text Add to dashboard Cite

show abstract

“…We have σ ̺,1 (X 1,2 ) = −X 3,4 and σ ̺,1 (X 1,3 ) = X 2,4 . Hence we can write C = C 0 + C 1 λ, with C 0 = a(X 1,2 − X 3,4 ) + b(X 1,3 + X 2,4 ), C 1 = cX 1,4 for some meromorphic functions a, b, c on S 2 . The harmonicity equations impose that ab ′ − ba ′ = 0, which means that b = αa for some constant α ∈ C. Hence given arbitrary meromorphic functions a, c on S 2 and a complex constant α,…”

Section: 31mentioning

confidence: 99%

Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations

Correia

Pacheco

2016

manuscripta math.

Self Cite

View full text Add to dashboard Cite

Abstract. Making use of Murakami's classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on S 2 .

show abstract

“…The simple connected Lie group Spin (7), seen as a subgroup of SO(8) via spinor representation, is given, up to conjugation, by [15] for details). Hence, the Grassmannian model of ΩSpin (7) with respect to the spinor representation is given by the following proposition, whose proof we omit since it is completely similar to that of Proposition 3.2 in [9]. Fix on ∧ • U, where U = span{e 1 , e 2 , e 3 }, the vector basis…”

Section: Canonical Elements Of Classical Lie Algebrasmentioning

confidence: 99%

Harmonic maps of finite uniton number and their canonical elements

Correia

Pacheco

2014

Ann Glob Anal Geom

View full text Add to dashboard Cite

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group G, whether G has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group Ω alg G of algebraic loops in G and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low dimensional spin groups making use of spinor representations.

show abstract

Harmonic maps of finite uniton number into G 2

Cited by 7 publications

References 12 publications

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations

Harmonic maps of finite uniton number and their canonical elements

Contact Info

Product

Resources

About