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

Abstract

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