Exceptional minimal surfaces in spheres

Abstract

We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy \cite{ET0}, 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 Kähler sphere $S^6$. Moreover, we study superconformal surfaces in odd dimensional spheres that are isometric to their polar and show a relation to pseudoholomorphic curves in $S^6$