Abstract
We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy (Rend Semin Mat Univ Padova, 79:185–202, 1998), we obtain a description of exceptional surfaces in terms of a set of absolute value type functions, the a-invariants, that determine the geometry of the higher order curvature ellipses and satisfy certain Ricci-type conditions. We show that the a-invariants determine these surfaces up to a multiparameter family of isometric minimal deformations, where the number of the parameters is precisely the number of non-vanishing Hopf differentials. We give applications to superconformal surfaces and pseudoholomorphic curves in the nearly Kahler sphere $${\mathbb{S}^{6}}$$ . Moreover, we study superconformal surfaces in odd dimensional spheres that are isometric to their polar and show a relation to pseudoholomorphic curves in $${\mathbb{S}^{6}}$$ .
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