# New constructions of twistor lifts for harmonic maps

Martin Svensson, John C. Wood

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.
Original language English Manuscripta Mathematica 144 3-4 457-502 0025-2611 https://doi.org/10.1007/s00229-014-0659-9 Published - 2014

Twistors
Harmonic Maps
Symmetric Spaces
Riemann Surface
Explicit Formula
If and only if

### Keywords

• math.DG
• 53C43 (Primary) 58E20 (Secondary)

### Cite this

Svensson, Martin ; C. Wood, John. / New constructions of twistor lifts for harmonic maps. In: Manuscripta Mathematica. 2014 ; Vol. 144, No. 3-4. pp. 457-502.
@article{4cdacd98691f40459f6dcb051246552d,
title = "New constructions of twistor lifts for harmonic maps",
abstract = "We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.",
keywords = "math.DG, 53C43 (Primary) 58E20 (Secondary)",
author = "Martin Svensson and {C. Wood}, John",
year = "2014",
doi = "10.1007/s00229-014-0659-9",
language = "English",
volume = "144",
pages = "457--502",
journal = "Manuscripta Mathematica",
issn = "0025-2611",
publisher = "Heinemann",
number = "3-4",

}

In: Manuscripta Mathematica, Vol. 144, No. 3-4, 2014, p. 457-502.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - New constructions of twistor lifts for harmonic maps

AU - Svensson, Martin

AU - C. Wood, John

PY - 2014

Y1 - 2014

N2 - We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.

AB - We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.

KW - math.DG

KW - 53C43 (Primary) 58E20 (Secondary)

U2 - 10.1007/s00229-014-0659-9

DO - 10.1007/s00229-014-0659-9

M3 - Journal article

VL - 144

SP - 457

EP - 502

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 3-4

ER -