An extremal class of 3-dimensional elliptic affine spheres (Q5937614)

The abundance of affine spheres dwarfs any attempt at a complete classification, and even the classification of subclasses is quite difficult. The authors give a complete classification of such a subclass, namely of all 3-dimensional proper elliptic affine spheres satisfying a particular intrinsic curvature condition (called Chen's equality). The authors show that these affine spheres admit a preferred ruling; thus their geometry is determined by 2-dimensional submanifolds transverse to this ruling. This fact is used to reduce the classification problem to the study of minimal surfaces in the 5-dimensional pseudo-sphere of index 3: \(S^5_3\subset\mathbb{R}^6\), where the minimal surfaces have the property that their ellipses of curvature are non-degenerate circles.