Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 457–502 | Cite as

New constructions of twistor lifts for harmonic maps

  • Martin SvenssonEmail author
  • John C. Wood


We show that given a harmonic map φ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a J 2-holomorphic twistor lift of φ (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.

Mathematics Subject Classification (2000)

53C43 58E20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bahy-El-Dien A., Wood J.C.: The explicit construction of all harmonic two-spheres in G 2(R n). J. Reine u. Angew. Math. 398, 36–66 (1989)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bahy-El-Dien A., Wood J.C.: The explicit construction of all harmonic two-spheres in quaternionic projective spaces. Proc. Lond. Math. Soc. 62(3), 202–224 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bolton J., Pedit F., Woodward L.M.: Minimal surfaces and the affine Toda field model. J. Reine Angew. Math. 459, 119–150 (1995)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bolton, J., Woodward, L.M.: The affine Toda equations and minimal surfaces. In: Harmonic maps and integrable systems, pp. 59–82. Vieweg, Braunschweig (1994). See for a downloadable version
  5. 5.
    Burstall F.E.: A twistor description of harmonic maps of a 2-sphere into a Grassmannian. Math. Ann. 274, 61–74 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Burstall F.E., Guest M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309, 541–572 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Burstall, F.E., Rawnsley, J.H.: Twistor Theory for Riemannian Symmetric Spaces. Lecture Notes in Mathematics, 1424. Springer, Berlin, Heidelberg (1990)Google Scholar
  8. 8.
    Burstall F.E., Salamon S.M.: Tournaments, flags, and harmonic maps. Math. Ann. 277, 249–265 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Burstall F.E., Wood J.C.: The construction of harmonic maps into complex Grassmannians. J. Diff. Geom. 23, 255–298 (1986)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Dai B., Terng C.-L.: Bäcklund transformations, Ward solitons, and unitons. J. Diff. Geom. 75, 57–108 (2007)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Davidov, J., Sergeev, A.G.: (1993) Twistor spaces and harmonic maps (Russian). Uspekhi Mat. Nauk 48, No. 3(291), 3–96 (1993); translation in Russian Math. Surveys 48(3), 1–91 (1993)Google Scholar
  12. 12.
    Dong Y., Shen Y.: Factorization and uniton numbers for harmonic maps into the unitary group U(n). Sci. China Ser. A 39, 589–597 (1996)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Dorfmeister J., Eschenburg J.-H.: Pluriharmonic maps, loop groups and twistor theory. Ann. Global Anal. Geom. 24, 301–321 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Eells J., Lemaire L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Eells, J., Salamon, S.: Twistorial construction of harmonic maps of surfaces into four-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), 589–640 (1986)Google Scholar
  16. 16.
    Eells J., Wood J.C.: Harmonic maps from surfaces to complex projective spaces. Adv. Math. 49, 217–263 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Erdem S., Wood J.C.: On the constructions of harmonic maps into a Grassmannian. J. Lond. Math. Soc. 28(2), 161–174 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Eschenburg J.-H., Tribuzy R.: Associated families of pluriharmonic maps and isotropy. Manuscripta Math. 95, 295–310 (1998)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Ferreira M.J., Simões B.A.: Explicit construction of harmonic two-spheres into the complex Grassmannian. Math. Z. 272, 151–174 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Ferreira M.J., Simões B.A., Wood J.C.: All harmonic 2-spheres in the unitary group, completely explicitly. Math. Z. 266, 953–978 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Guest M.A.: Harmonic maps, loop groups, and integrable systems London Mathematical Society Student Texts 38. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  22. 22.
    Guest, M.A.: An update on harmonic maps of finite uniton number, via the zero curvature equation. Integrable systems, topology, and physics (Tokyo, 2000), Contemp. Math. 309, pp. 85–113. Am. Math. Soc., Providence, RI (2002)Google Scholar
  23. 23.
    He Q., Shen Y.B.: Explicit construction for harmonic surfaces in U(N) via adding unitons. Chin. Ann. Math. Ser. B 25, 119–128 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Koszul J.L., Malgrange B.: Sur certaines structures fibrées complexes. Arch. Math. 9, 102–109 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Pacheco R.: Harmonic two-spheres in the symplectic group Sp(n). Int. J. Math. 17, 295–311 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Pacheco R.: On harmonic tori in compact rank one symmetric spaces. Differ. Geom. Appl. 27, 352–361 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford (1986)Google Scholar
  28. 28.
    Ramanathan J.: Harmonic maps from S 2 to G 2, 4. J. Differ. Geom. 19, 207–219 (1984)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Rawnsley, J.: f-structures, f-twistor spaces and harmonic maps, Geometry seminar “Luigi Bianchi” II–1984, pp. 85–159, Lecture Notes in Math., 1164. Springer, Berlin (1985)Google Scholar
  30. 30.
    Salamon S.: Quaternionic Kähler manifolds. Invent. Math. 67, 143–171 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Salamon, S.: Harmonic and Holomorphic Maps. Geometry Seminar “Luigi Bianchi” II–1984, pp. 161–224, Lecture Notes in Math., 1164. Springer, Berlin (1985)Google Scholar
  32. 32.
    Segal, G.: Loop Groups and Harmonic Maps, Advances in Homotopy Theory (Cortona, 1988), pp. 153–164. London Math. Soc. Lecture Notes Ser., 139. Cambridge University Press, CambridgeGoogle Scholar
  33. 33.
    Svensson M., Wood J.C.: Filtrations, factorizations and explicit formulae for harmonic maps. Commun. Math. Phys. 310, 99–134 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Uhlenbeck K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differ. Geom. 30, 1–50 (1989)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Wolfson J.G.: Harmonic sequences and harmonic maps of surfaces into complex Grassmann manifolds. J. Differ. Geom. 27, 161–178 (1988)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Wood J.C.: Explicit construction and parametrization of harmonic two-spheres in the unitary group. Proc. Lond. Math. Soc. 58(3), 608–624 (1989)CrossRefzbMATHGoogle Scholar
  37. 37.
    Wood, J.C.: Explicit constructions of harmonic maps. In: Loubeau E., Montaldo S. (eds.) Harmonic Maps and Differential Geometry, pp. 41–74. Contemp. Math., 542. Am. Math. Soc. (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, CP3–Origins Centre of Excellence for Particle Physics and PhenomenologyUniversity of Southern DenmarkOdense MDenmark
  2. 2.Department of Pure MathematicsUniversity of LeedsLeedsUK

Personalised recommendations