A family of higher genus complete minimal surfaces that includes the Costa-Hoffman-Meeks one

2023/03/24 by Irene I. Onnis, Onnis, Irene I., Bárbara Corominas Valério +3
Mathematics · #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2303.13751

openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean 3-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another interesting property is that the symmetry group of the genus k surfaces Σ_k,x is the dihedral group with 4(k+1) elements. Moreover, in particular, for |x|=1 we find the family of the Costa-Hoffman-Meeks embedded minimal surfaces, which have two catenoidal ends and a middle flat end.

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