Self-shrinkers with any number of ends in ℝ³ by stacking ℝ²

2025/07/24 by Shao, Guanhua, Jiahua Zou, Zou, Jiahua
Mathematics · #53A10 (Primary) #53E10 (Secondary) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2507.18825

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

Abstract

For each half-integer J and large enough integer m we construct by PDE gluing methods a self-shrinker \breveM[J,m] with 2J+1 ends and genus 2J(m-1). \breveM[J,m] resembles the stacking of 2J+1 levels of the plane ℝ² in ℝ³ that have been connected by 2Jm catenoidal bridges with m bridges connecting each pair of adjacent levels. It observes the symmetry of an m-gonal prism (when J is a half integer) or an m-gonal antiprism (when J is an integer). The construction is based on the Linearised Doubling (LD) methodology which was first introduced by Kapouleas in the construction of minimal surface doublings of \mathbbS²_eq in \mathbbS³.

Citations

• Generating Shrinkers by Mean Curvature Flow
• Fattening in mean curvature flow
• Closed mean curvature flows with asymptotically conical singularities
• Disc stackings and their Morse index
• Free-boundary minimal surfaces with connected boundary in the 3-ball by tripling the equatorial disc
• Self-shrinking Platonic solids
• Free boundary minimal surfaces in the unit 3-ball
• Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I
• Tridiagonal Toeplitz matrices: properties and novel applications
• Stability and compactness for complete f-minimal surfaces
• Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends

Related

• Self-shrinking Platonic solids
• Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers
• Self-shrinkers whose asymptotic cones fatten
• Closed self-shrinking surfaces in ℝ³ via the torus
• Immersed self-shrinkers