Superconformal harmonic surfaces in De Sitter space-times

We consider harmonic maps of Riemann surfaces in De Sitter space-times S1n,n≥3 with maximal isotropy dimension, also called superconformal. Harmonic sequences are constructed for these maps which are used to study their geometry. Global properties such as (linear) fullness and rigidity are discussed and polar maps of superconformal harmonic into odd-dimensional De Sitter space-times are studied. Lastly a characterization of superconformal minimal immersed tori is obtained generalizing a result by Sakaki [M. Sakaki, Space-like minimal surfaces in four-dimensional Lorentzian space forms, Tsukuba J. Math. 25 (2) (2001) 239-246].