arXiv subjects
Publications and source records attributed to Maria João Ferreira.
We find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group O(n) in terms of free holomorphic data which lead to formulae for all such harmonic maps. Our work reveals an interesting correspondence between certain harmonic maps and the free Weierstrass representation of null curves and minimal surfaces in 3- and 4-space.
In [5], together with J. C. Wood, the authors gave a completely explicit formula for all harmonic maps from $2$-spheres to the unitary group $U(n)$ in terms of freely chosen meromorphic functions on $S^2$. The simplest harmonic maps are the isotropic ones. Using Morse theory Burstall and Guest [1] showed that the harmonic maps come in clusters labeled by the isotropic ones. In this work, using the formula for harmonic maps aforementioned, we describe explicitly this procedure, showing how all harmonic maps can be built from the isotropic ones. [1] F.~E.\ Burstall and M.~A.\ Guest, \textit{Harmonic two-spheres in compact symmetric spaces, revisited}, Math. Ann. 309 (1997) 541--572. [5] M.~J. Ferreira, B.~A Simões and J.~C. Wood \emph{All harmonic $2$-spheres in the unitary group, completely explicitly}, Math. Z. {\bf 266} (2010), 953--978.
We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual dbar-problems or loop group factorizations. We interpret our constructions using Segal's Grassmannian model, giving an explicit factorization of the algebraic loop group, and showing how to obtain harmonic maps into a Grassmannian.