The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1132–1156 | Cite as

Minimal Surfaces in \(\mathbb{S}^{2} \times\mathbb{S}^{2}\)

  • Francisco Torralbo
  • Francisco UrbanoEmail author


A general study of minimal surfaces of the Riemannian product of two spheres \(\mathbb {S}^{2}\times \mathbb {S}^{2}\) is tackled. We establish a local correspondence between (non-complex) minimal surfaces of \(\mathbb {S}^{2} \times \mathbb {S}^{2}\) and a certain pair of minimal surfaces of the sphere \(\mathbb {S}^{3}\). This correspondence also allows us to link minimal surfaces in \(\mathbb{S}^{3}\) and in the Riemannian product \(\mathbb {S}^{2} \times \mathbb {R}\). Some rigidity results for compact minimal surfaces are also obtained.


Surfaces Minimal Complex surfaces 

Mathematics Subject Classification

53C42 53C40 


  1. 1.
    Asperti, A.C., Ferus, D., Rodríguez, L.: Surfaces with nonzero normal curvature. Rend. Sci. Fis. Mat. Lincei 73, 109–115 (1982) Google Scholar
  2. 2.
    Bryant, R.: Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Differ. Geom. 17, 455–473 (1982) zbMATHGoogle Scholar
  3. 3.
    Castro, I., Urbano, F.: Minimal Lagrangian surfaces in \(\mathbb {S}^{2}\times \mathbb {S}^{2}\). Commun. Anal. Geom. 15, 217–248 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51, 43–55 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen, B.Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces I. Duke Math. J. 44, 745–755 (1997) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Eschenburg, J.-T., Guadalupe, I.V., Tribuzy, R.: The fundamental equations of minimal surfaces in \(\mathbb{C}P^{2}\). Math. Ann. 270, 571–598 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hauswirth, L., Kilian, M., Schmidt, M.U.: Finite type minimal annuli in \(\mathbb{S}^{2} \times\mathbb{R}\). arXiv:1210.5606v1 [math.DG]
  8. 8.
    Hitchin, N.J.: Harmonic maps from a 2-torus to the 3-sphere. J. Differ. Geom. 31, 627–710 (1990) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Lawson, H.B.: Complete minimal surfaces in \(\mathbb {S}^{3}\). Ann. Math. 92, 335–374 (1970) CrossRefzbMATHGoogle Scholar
  10. 10.
    Li, P., Yau, S.-T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69, 269–291 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Micallef, M.J., Wolfson, J.G.: The second variation of area of minimal surfaces in four-manifolds. Math. Ann. 295, 245–267 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Montiel, S., Urbano, F.: Second variation of superminimal surfaces into self-dual Einstein four-manifolds. Trans. Am. Math. Soc. 349, 2253–2269 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569–573 (1970) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Schoen, R., Yau, S.T.: On univalent harmonic maps between surfaces. Invent. Math. 44, 265–278 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Torralbo, F., Urbano, F.: Surfaces with parallel mean curvature vector in \(\mathbb {S}^{2} \times \mathbb {S}^{2}\) and \(\mathbb {H}^{2} \times \mathbb {H}^{2}\). Trans. Am. Math. Soc. 364, 785–813 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Torralbo, F., Urbano, F.: Compact stable minimal submanifolds. To appear in Proc. Am. Math. Soc. Google Scholar
  17. 17.
    Wolfson, J.G.: Minimal surfaces in Kähler surfaces and Ricci curvature. J. Differ. Geom. 29, 281–294 (1989) zbMATHMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

Personalised recommendations