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

Abstract

A sequence of constant mean curvature surfaces $Σ_j$ with mean curvature $H_j \to \infty$ 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 $j \in \N$. 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 $Γ$.