2011/08/30 by Vlachos, Theodoros
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 53A10 #Secondary 53C42
paper · doi:10.48550/arxiv.1108.5834
We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy \citeET0, 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 S6. Moreover, we study superconformal surfaces in odd dimensional spheres that are isometric to their polar and show a relation to pseudoholomorphic curves in S6