Abstract
Making use of Murakami’s classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on \(S^2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahy-El-Dien, A., Wood, J.C.: The explicit construction of all harmonic two-spheres in \(G_2({\mathbb{R}}^n)\). J. Reine Angew. Math. 398, 36–66 (1989)
Burstall, F.E., Guest, M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309(4), 541–572 (1997)
Burstall, F.E., Rawnsley, J.H.: Twistor Theory for Riemannian Symmetric Spaces. In: Dols, A., Eckmann, B., Takens, F. (eds.) Lecture Notes in Mathematics, vol. 1424. Springer, Berlin, Heidelberg, New York (1990)
Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Diff. Geom. 1, 111–125 (1967)
Correia, N., Pacheco, R.: Harmonic maps of finite uniton number into \(G_2\). Math. Z. 271(1–2), 13–32 (2012)
Correia, N., Pacheco, R.: Extended Solutions of the Harmonic Map Equation in the Special Unitary Group. Q. J. Math. 65(2), 637–654 (2014)
Correia, N., Pacheco, R.: Harmonic maps of finite uniton number and their canonical elements. Ann. Global Anal. Geom. 47(4), 335–358 (2015)
Dorfmeister, J., Pedit, F., Wu, H.: Weiestrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6, 633–668 (1998)
Eschenburg, J.-H., Mare, A.-L., Quast, P.: Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers. Bull. London Math. Soc. 42(6), 1121–1133 (2010)
Eells, J., Wood, J.C.: Harmonic maps from surfaces to complex projective spaces. Adv. in Math. 49(3), 217–263 (1983)
Fulton, W., Harris, J.: Representation theory. A first course. In: Axler, S., Gehring, F.W., Ribet, K.A. (eds.) Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)
Guest, M.A., Ohnita, Y.: Loop group actions on harmonic maps and their applications. Harmonic maps and integrable systems, 273–292, Aspects Math., E23, Vieweg, Braunschweig (1994)
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978)
Ma, H.: Explicit construction of harmonic two-spheres in \(SU(3)/SO(3)\). Kyushu J. Math. 55, 237–247 (2001)
Murakami, S.: Sur la classification des algèbres de Lie réelles et simples. Osaka J. Math. 2, 291–307 (1965)
Pressley, A.N., Segal, G.B.: Loop Groups. Oxford University Press, (1986)
Uhlenbeck, K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30, 1–50 (1989)
Ziller, W.: Lie groups: representation theory and symmetric spaces. Notes for a course given in the fall of 2010 at the University of Pennsylvania and 2012 at IMPA. www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf. Accessed 20May 2015
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Correia, N., Pacheco, R. Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations. manuscripta math. 152, 399–432 (2017). https://doi.org/10.1007/s00229-016-0862-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-016-0862-y