On the Canham Problem: Bending Energy Minimizers for any Genus and Isoperimetric Ratio

2021/04/20 by Robert Kusner, Kusner, Robert, Peter McGrath +1 · 2 citations
Mathematics · #49Q10 #53A10 #53C42 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Optimization and Control (math.OC) #Soft Condensed Matter (cond-mat.soft)

paper · pdf · doi:10.48550/arxiv.2104.10045

openalex publication_date 2021/04/20 · openalex created_date 2023/01/22 · openalex updated_date 2026/07/28

Abstract

Building on work of Mondino-Scharrer, we show that among closed, smoothly embedded surfaces in ℝ³ of genus g and given isoperimetric ratio v, there exists one with minimum bending energy W. We do this by gluing g+1 small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation Δ(Δ+2)φ=0 on the (g+1)-punctured sphere \mathbbS² to construct a comparison surface of genus g with arbitrarily small isoperimetric ratio v∈ (0, 1) and W < 8π.

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