Isothermic Surfaces Obtained From Harmonic Maps in $S^6$

Rui Pacheco

Abstract: The harmonicity of a smooth map from a Riemann surface into the 6-dimensional sphere $S^6$ amounts to the closeness of a certain 1-form that can be written in terms of the nearly Kähler structure of $S^6$. We will prove that the immersions $F$ in ${ℝ}^7$ obtained from superconformal harmonic maps in $S^3\subset S^6$ 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 ${ℝ}^3$ that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in ${ℝ}^7$.

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

Classification (MSC2000): 53C43,53C42, 53A10,53A07

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