Minimal surfaces in spheres and a Ricci-like condition

TL;DR

This paper characterizes minimal surfaces in spheres that are locally isometric to pseudoholomorphic curves in $ ext{S}^5$, establishing a Ricci-like condition and classifying such surfaces as flat or direct sums of pseudoholomorphic curve surfaces.

Contribution

It introduces a Ricci-like condition characterizing these minimal surfaces and reduces their classification to that of exceptional surfaces related to pseudoholomorphic curves.

Findings

Minimal surfaces satisfying the Ricci-like condition are exceptional surfaces.

Such surfaces are either flat or direct sums of pseudoholomorphic curve surfaces.

The classification of these surfaces is achieved under global assumptions.

Abstract

We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic $\mathbb{S}^{5}$ in the nearly K{\"a}hler sphere $\mathbb{S}^6$. Being locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$ turns out to be equivalent to the Ricci-like condition $\Delta\log(1-K)=6K,$ where $K$ is the Gaussian curvature of the induced metric. Besides flat minimal surfaces in spheres, direct sums of surfaces in the associated family of pseudoholomorphic curves in $\mathbb{S}^5$ do satisfy this Ricci-like condition. Surfaces in both classes are exceptional surfaces. These are minimal surfaces whose all Hopf differentials are holomorphic, or equivalently the curvature ellipses have constant eccentricity up to the last but one. Under appropriate global assumptions, we prove that minimal surfaces in spheres that satisfy this Ricci-like condition are indeed exceptional. Thus, the classification of these surfaces is reduced to the classification of exceptional surfaces that are locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5.$ In fact, we prove, among other results, that such exceptional surfaces in odd dimensional spheres are flat or direct sums of surfaces in the associated family of a pseudoholomorphic curve in $\mathbb{S}^5$.