On Large Volume Preserving Stable CMC Surfaces in Initial Data Sets

Abstract

Let $(M, g)$ be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset $K \subset M$, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid $K$. This complements the results of G. Huisken and S.-T. Yau and of J. Qing and G. Tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass $m \gt 0$. The analysis in G. Huisken and S.-T. Yau and in J. Qing and G. Tian takes place in the asymptotic regime of $M$. Here we adapt ideas from the minimal surface proof of the positive mass theorem by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.