Minimal surfaces with reflectionally symetric sequences

Abstract

We study minimal surfaces in an odd dimensional unit sphere S²ⁿ⁺¹ for which sufficiently many higher-order ellipses of curvature are circles. In previous papers, see Anti´c (preprint) [1] and Bolton and Vrancken (to appear) [3], we associated with one such minimal surface a whole sequence of minimal surfaces in S²ⁿ⁺¹, and we instigated the investigation of the geometric properties of this sequence. In particular, we studied those surfaces for which the sequence has reflectional symmetry. This symmetry may be reflection in one of the minimal surfaces or reflection in the middle point of two adjacent surfaces. The first case was studied in [1] and [3], where it was shown that the minimal surface in question is not linearly full, and the second case is investigated here.