An extremal class of 3-dimensional elliptic affine spheres

Abstract

In analogy to an inequality of Chen [Che93], Scharlach, Simon, Verstraelen and Vrancken [SSVV97] have found a new inequality for (equi-) affine spheres. This inequality is optimal and in this paper we classify those 3-dimensional elliptic affine spheres for which the corresponding equality is assumed. This is achieved through reducing the problem to the problem of classifying those 2-dimensional minimal surfaces in $S_{3}^{5}$ whose ellipses of curvature are circles. We end with the investigation of 2-dimensional minimal surfaces in $S_{3}^{5}$ with positive definite induced metric whose ellipses of curvature are circles.