# Harmonic maps of finite uniton number into G2

@article{Correia2010HarmonicMO, title={Harmonic maps of finite uniton number into G2}, author={N. Correia and Rui Pacheco}, journal={Mathematische Zeitschrift}, year={2010}, volume={271}, pages={13-32} }

We establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional Lie group G2 in terms of the Grassmannian model for the group of based algebraic loops in G2. A description of the “Frenet frame data” for such harmonic maps is given. In particular, we show that harmonic spheres into G2 correspond to solutions of certain algebraic systems of quadratic and cubic equations.

## 9 Citations

Harmonic surfaces in the Cayley plane

- MathematicsJournal of the London Mathematical Society
- 2020

We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane OP2=F4/Spin(9) . By exhibiting this symmetric space as a submanifold of the Grassmannian of…

Harmonic maps into the exceptional symmetric space G2/SO(4)

- MathematicsJ. Lond. Math. Soc.
- 2015

We show that a harmonic map from a Riemann surface into the exceptional symmetric space G2/ SO (4) has a J2 ‐holomorphic twistor lift into one of the three flag manifolds of G2 if and only if it is…

Harmonic maps of finite uniton number and their canonical elements

- Mathematics
- 2014

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $$G$$G, whether $$G$$G has trivial centre or not, in terms of certain pieces…

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

- Mathematics
- 2014

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…

Willmore Surfaces in Spheres via Loop Groups IV: On Totally Isotropic Willmore Two-Spheres in S6

- Mathematics
- 2014

By using Wu’s formula and a straightforward computations of some special Laurant polynomials in the loop group � − SO(1,7,C)σ, we derive a description of totally isotropic Willmore two-spheres in S 6…

Willmore surfaces in $S^{n+2}$ by the loop group method II: surfaces of finite uniton type

- Mathematics
- 2013

The main goal of this paper is to give a classification of Willmore two-spheres in S6, or in general in Sn+2, by use of the theory developed by Burstall and Guest for harmonic maps of finite uniton…

Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials

- Mathematics
- 2014

This paper is to provide a coarse classification of Willmore two-spheres in $S^{n+2}$, and to give explicit applications to the case of $S^6$. We use the theory developed by Burstall and Guest for…

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

- Mathematics
- 2014

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…

## References

SHOWING 1-10 OF 11 REFERENCES

Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

- Mathematics
- 2009

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere,…

SINGULAR DRESSING ACTIONS ON HARMONIC MAPS

- Mathematics
- 2011

In this paper we prove that any harmonic map φ from a two-sphere S into an arbitrary compact semisimple matrix Lie group G may be reduced to a constant by using the singular dressing actions…

Harmonic two-spheres in compact symmetric spaces, revisited

- Mathematics
- 1996

Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new…

An update on harmonic maps of finite uniton number, via the zero curvature equation

- Mathematics
- 2001

This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck.…

HARMONIC TWO-SPHERES IN THE SYMPLECTIC GROUP Sp(n)

- Mathematics
- 2006

We shall exploit the Grassmannian theoretic point of view introduced by Segal in order to study harmonic maps from a two-sphere into the symplectic group Sp(n). By using this methodology, we shall be…

Representation Theory: A First Course

- Mathematics
- 1991

This volume represents a series of lectures which aims to introduce the beginner to the finite dimensional representations of Lie groups and Lie algebras. Following an introduction to representation…

Harmonic maps into Lie groups: classical solutions of the chiral model

- Mathematics
- 1989

On considere deux aspects de la structure algebrique de #7B-M, l'espace des applications harmoniques d'un domaine a 2 dimensions simplement connexe dans un groupe de Lie reel G R , la forme reelle…

Loop groups and harmonic maps, Advances in homotopy theory

- London Math. Soc. Lecture notes 139,
- 1989