CMC Surfaces in Riemannian Manifolds Condensing to a Compact Network of Curves

2009/10/23 by Adrian Butscher, Butscher, Adrian
Mathematics · #53Cxx #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG #msc:53Cxx

paper · pdf · doi:10.48550/arxiv.0910.4442

26 pages

arxiv created 2009/10/23 · openalex publication_date 2009/10/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A sequence of constant mean curvature surfaces Σ_j with mean curvature H_j → ∞ in a three-dimensional manifold M condenses to a compact and connected graph Γ consisting of a finite union of curves if Σ_j is contained in a tubular neighbourhood of Γ of size \mathcal O(1/H_j) for every j ∈ \N. This paper gives sufficient conditions on Γ for the existence of a sequence of compact, embedded constant mean curvature surfaces condensing to Γ. The conditions are: each curve in γ is a critical point of a functional involving the scalar curvature of M along γ; and each curve must satisfy certain regularity, non-degeneracy and boundary conditions. When these conditions are satisfied, the surfaces Σ_j can be constructed by gluing together small spheres of radius 2/H_j positioned end-to-end along the edges of Γ.

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