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Isothermic surfaces obtained from harmonic maps in S6

Pacheco Rui (Universidade da Beira Interior, Centro de Matemática e Aplicações (CMA-UBI), Covilhã, Portugal)

The harmonicity of a smooth map from a Riemann surface into the 6-dimensional sphere S6 amounts to the closeness of a certain 1-form that can be written in terms of the nearly Kähler structure of S6. We will prove that the immersions F in R7 obtained from superconformal harmonic maps in S3  S6 by integration of the corresponding closed 1-forms are isothermic. The isothermic surfaces so obtained include a certain class of constant mean curvature surfaces in R3 that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in R7.

Keywords: Harmonic maps, isothermic surfaces, parallel mean curvature, pseudo-umbilical surfaces, seven dimensional cross product