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Any Grassmannian can be embedded in a unitary group U(n). In Sect. 3, we see how Gauss transforms are a special case of K. Uhlenbeck’s construction [Uh] of adding a uniton, also called flag transform, which provides a method of modifying harmonic maps from a surface to the unitary group to get new ones. This process gives all harmonic 2-spheres in the unitary group as a ‘product’ of unitons. We 1991 Mathematics Subject Classification. Primary 53C42; Secondary 58E20. Key words and phrases. Differential geometry, harmonic map, uniton, Grassmannian model, loop group, nonlinear sigma model.

1. Harmonic maps between Riemannian manifolds. Let (M, g) and (N, h) be smooth Riemannian manifolds and let ϕ : (M, g) → (N, h) be a smooth map between them. The energy or Dirichlet integral of ϕ over a compact domain D of M is defined by E(ϕ, D) = 1 2 D |dϕ|2 ωg EXPLICIT CONSTRUCTIONS OF HARMONIC MAPS 43 3 where ωg = volume measure and, for any p ∈ M , |dϕp |2 is the Hilbert–Schmidt square norm of the differential dϕp of ϕ at p. In local coordinates, this is given by β |dϕp |2 = g ij hαβ ϕα i ϕj .

Math. 93 (1988), no. 3, 557–607. odinger 13. P. G. Grinevich and M. U. Schmidt, Closed curves in R3 and the nonlinear Schr¨ equation, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEDS ’79 (Gallipoli, 1999), World Sci. , River Edge, NJ, 2000, pp. 139–145. 14. K. Grosse-Brauckmann, R. Kusner, and J. Sullivan, Coplanar constant mean curvature surfaces, Comm. Anal. Geom. 15 (2007), no. 5, 985–1023. 15. K. Große-Brauckmann, R. B. Kusner, and J. M. Sullivan, Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero, J.

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A C^2 Finite Element and Interpolation by Gao J.


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