Explicit construction of harmonic two-spheres into the complex Grassmannian


We present an explicit description of all harmonic maps of finite uniton number from a Riemann surface into a complex Grassmannian. Namely, starting from a constant map Q and a collection of meromorphic functions and their derivatives, we show how to algebraically construct all harmonic maps from the two-sphere into a given Grassmannian \({G_p(\mathbb C^n)}\) . In this setting the uniton number depends on Q and p and we obtain a sharp estimate for it.

This is a preview of subscription content, log in to check access.


  1. 1

    Burstall F.E., Guest M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309, 541–572 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2

    Burstall F.E., Wood J.C.: The construction of harmonic maps into complex Grassmannians. J. Diff. Geom. 23, 255–298 (1986)

    MathSciNet  MATH  Google Scholar 

  3. 3

    Chern S.S., Wolfson J.G.: Harmonic maps of S 2 into a complex Grassmann manifold. J. Proc. Nat. Acad. Sci. 82, 2217–2219 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

    Chern S.S., Wolfson J.G.: Harmonic maps of the two-sphere into a complex Grassmann manifold II. J. Ann. of Math. 125, 301–335 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5

    Dai B., Terng C.-L.: Bäcklund transformations, Ward solitons, and unitons. J. Differ. Geom. 75, 57–108 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6

    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)

    MathSciNet  MATH  Google Scholar 

  7. 7

    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)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8

    Guest M.A.: Harmonic maps, loop groups, and integrable systems, vol. 38. London Mathematical Society Student Texts, London (1997)

    Google Scholar 

  9. 9

    Guest, M.A.: An update on harmonic maps of finite uniton number, via the zero curvature equation. In: Integrable Systems, Topology, and Physics (Tokyo, 2000), vol. 309. pp. 85–113. Contemp. Math., American Mathematical Society, Providence (2002)

  10. 10

    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)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11

    Koszul J.L., Malgrange B.: Sur certaines structures fibrées complexes. Arch. Math. 9, 102– 109 (1958)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

    Pressley A., Segal G.: Loop groups, Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford (1986)

    Google Scholar 

  13. 13

    Segal, G.: Loop groups and harmonic maps. In: Advances in Homotopy Theory (Cortona, 1988), pp. 153–164, vol. 139. London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge (1989)

  14. 14

    Uhlenbeck K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differ. Geom. 30, 1–50 (1989)

    MathSciNet  MATH  Google Scholar 

  15. 15

    Wood J.C.: Explicit construction and parametrization of harmonic two-spheres in the unitary group. Proc. Lond. Math. Soc. 58(3), 608–624 (1989)

    MATH  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Bruno Ascenso Simões.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ferreira, M.J., Simões, B.A. Explicit construction of harmonic two-spheres into the complex Grassmannian. Math. Z. 272, 151–174 (2012). https://doi.org/10.1007/s00209-011-0927-2

Download citation


  • Harmonic map
  • Uniton
  • Grassmannian
  • Loop group

Mathematics Subject Classification (2000)

  • Primary 58E20
  • Secondary 53C43