Closed self-shrinking surfaces in ℝ³ via the torus

2011/11/30 by Niels Martin Møller, Møller, Niels Martin · 2 citations
Mathematics · #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.1111.7318

openalex publication_date 2011/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct many closed, embedded mean curvature self-shrinking surfaces Σ_g²⊆ℝ³ of high genus g=2k, k∈ ℕ. Each of these shrinking solitons has isometry group equal to the dihedral group on 2g elements, and comes from the "gluing", i.e. desingularizing of the singular union, of the two known closed embedded self-shrinkers in ℝ³: The round 2-sphere \mathbbS², and Angenent's self-shrinking 2-torus \mathbbT² of revolution. This uses the results and methods N. Kapouleas developed for minimal surfaces in \citeKa97--\citeKa.

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