Narrow Results

In this paper, we introduce a nonlinear ODE method to construct constant mean curvature (CMC) surfaces in Riemannian manifolds with symmetry. As an application, we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}={(1+\frac{1}{l})}^2{\delta }_{ij}+O(l^{-2})$. The existence of unstable CMC spheres tells us that the stability condition in Qing–Tian’s work [On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds, J. Amer. Math. Soc.20(4) (2007) 1091–1110] cannot be removed generally.

In a graph $G:=(V(G),E(G))$, a subset $M$ of the vertex set $V(G)$ is a module (or interval, clan) of $G$ if every vertex outside $M$ is adjacent to all or none of $M$. The empty set, the singleton sets, and the full set of vertices are trivial modules. The graph $G$ is indecomposable (or prime) if all its modules are trivial. If $G$ is indecomposable, we say that an edge $e$ of $G$ is a removable edge if $G-e$ is indecomposable (here $G-e:=(V(G),E(G)-\{e\})$). The graph $G$ is said to be unstable if it has no removable edges. For a positive integer $k$, the $k$th power $G^k$ of a graph $G$ is the graph obtained from $G$ by adding an edge between all pairs of vertices of $G$ with distance at most $k$. A graph $G$ of order $n$ (i.e., $|V(G)|=n$) is said to be $p$-placeable into $G^k$, if $G^k$ contains $p$ edge-disjoint copies of $G$. In this paper, we answer a question, suggested by Boudabbous in a personal communication, concerning unstable graphs and packing into their fifth power as follows: First, we give a characterization of the unstable graphs which is deduced from the results given by Ehrenfeucht, Harju and Rozenberg (the theory of $2$-structures: a framework for decomposition and transformation of graphs). Second, we prove that every unstable graph $G$ is $2$-placeable into $G^5$.