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

Martin Svensson, John C. Wood

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is nilconformal', i.e., has nilpotent derivative. The class of nilconformal maps includes those of finite uniton number studied by N. Correia and R. Pacheco, however we exhibit examples which are not of finite uniton number. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces.
Original language English Journal of the London Mathematical Society 91 291-319 0024-6107 https://doi.org/10.1112/jlms/jdu073 Published - 2015

Harmonic Maps
Symmetric Spaces
Twistor Space
Twistors
Flag Manifold
Riemann Surface
If and only if
Derivative

• math.DG

### Cite this

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title = "Harmonic maps into the exceptional symmetric space G2/SO(4)",
abstract = "We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is nilconformal', i.e., has nilpotent derivative. The class of nilconformal maps includes those of finite uniton number studied by N. Correia and R. Pacheco, however we exhibit examples which are not of finite uniton number. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces.",
keywords = "math.DG",
author = "Martin Svensson and {C. Wood}, John",
note = "(Submitted on 28 Mar 2013 (v1), last revised 6 Sep 2013 (this version, v2))",
year = "2015",
doi = "10.1112/jlms/jdu073",
language = "English",
volume = "91",
pages = "291--319",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Heinemann",

}

In: Journal of the London Mathematical Society, Vol. 91, 2015, p. 291-319.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Harmonic maps into the exceptional symmetric space G2/SO(4)

AU - Svensson, Martin

AU - C. Wood, John

N1 - (Submitted on 28 Mar 2013 (v1), last revised 6 Sep 2013 (this version, v2))

PY - 2015

Y1 - 2015

N2 - We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is nilconformal', i.e., has nilpotent derivative. The class of nilconformal maps includes those of finite uniton number studied by N. Correia and R. Pacheco, however we exhibit examples which are not of finite uniton number. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces.

AB - We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is nilconformal', i.e., has nilpotent derivative. The class of nilconformal maps includes those of finite uniton number studied by N. Correia and R. Pacheco, however we exhibit examples which are not of finite uniton number. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces.

KW - math.DG

U2 - 10.1112/jlms/jdu073

DO - 10.1112/jlms/jdu073

M3 - Journal article

VL - 91

SP - 291

EP - 319

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

ER -