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 $\mathbb{S}^5$.

中文翻译：

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球体中的最小表面和类 Ricci 条件

我们处理球体中的最小表面，这些表面与近似 K{a}hler 球体 $\mathbb{S}^6$ 中的完全测地线 $\mathbb{S}^{5}$ 中的伪全纯曲线局部等距。与 $\mathbb{S}^5$ 中的伪全纯曲线局部等距结果证明等价于类 Ricci 条件 $\Delta\log(1-K)=6K,$ 其中 $K$ 是高斯曲率的诱导度量。除了球体中平坦的最小曲面之外，$\mathbb{S}^5$ 中相关的伪全纯曲线族中的曲面的直接和确实满足这种类 Ricci 条件。这两个类别中的曲面都是特殊曲面。这些是最小表面，其所有 Hopf 微分都是全纯的，或者等效地，曲率椭圆具有恒定的偏心率，直到最后一个。在适当的全局假设下，我们证明满足这种类 Ricci 条件的球体中的最小表面确实是特殊的。因此，这些曲面的分类被简化为局部等距于 $\mathbb{S}^5 中的伪全纯曲线的异常曲面的分类。维球体是 $\mathbb{S}^5$ 中伪全纯曲线相关族中的平面或直接和。