2012/09/17 by Óscar Perdomo, Perdomo, Oscar · 1 citation
Mathematics · #53C10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1209.3651
openalex publication_date 2012/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+√(1+H2)) there is an axially symmetry surface ΣH,C with mean curvature H. In 2010, Perdomo showed that for every H between cot(π/m) and (m2-2)/(2(m2-1)1/2), there exists an embedded axially symmetric example with non constant principal curvatures that is invariant under the ciclic group Zm. Andrews and Li, showed that these examples are the only non-isoparametric embedded examples in the family when H>0. In this paper we study those examples in the family with H<0. We prove that there are no embedded examples in the family when H<0 and we also prove that for every integer m>2 there is a properly immersed example in this family that contains a great circle and is invariant under the ciclic group Zm. We will say that these examples contain the axis of symmetry. Finally we show that every non-isoparametric surface ΣH,C is either properly immersed invariant under the ciclic group Zm for some integer m>1 or it is dense in the region bounded by two isoparametric tori if the surface ΣH,C does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface ΣH,C contains the axis of symmetry.