Exceptional minimal surfaces in spheres, pseudoholomorphic curves in $S^{6}$ and polar surfaces

Details Exceptional minimal surfaces in spheres, pseudoholomorphic curves in $S^{6}$ and polar surfaces Exceptional minimal surfaces in spheres, pseudoholomorphic curves in $S^{6}$ and polar surfaces

2011-08-30

arxiv.org/abs/1108.5834v1

Mathematics

Differential Geometry

Scientific paper

We deal with a class of minimal surfaces in spheres for which all Hopf differentials are holomorphic, the so called exceptional surfaces. We obtain a characterization of exceptional surfaces in any sphere in terms of a set of scalar invariants, the $a$-invariants, that satisfy certain Ricci-type conditions. We show that these surfaces are determined by the $a$-invariants 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 in particular to pseudoholomorphic curves in the nearly K\"{a}hler sphere $S^{6}$. Moreover, we study superconformal surfaces in odd dimensional spheres that are isometric to their polar and show how they are related to pseudoholomorphic curves in $S^{6}$.

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