Exceptional minimal surfaces in spheres

TL;DR

This paper characterizes exceptional minimal surfaces in spheres with holomorphic Hopf differentials using $a$-invariants, describing their geometry, deformations, and applications to special classes like superconformal and pseudoholomorphic curves.

Contribution

It extends previous results by providing a new description of exceptional minimal surfaces via $a$-invariants and explores their deformation space and applications.

Findings

$a$-invariants determine exceptional surfaces up to isometric deformations.

The number of parameters in deformations equals the number of non-vanishing Hopf differentials.

Applications include classifications of superconformal surfaces and pseudoholomorphic curves in $S^{6}$.

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 K\"{a}hler 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 $S^{6}$