Translator Disclaimer
May 2012 On Large Volume Preserving Stable CMC Surfaces in Initial Data Sets
Michael Eichmair, Jan Metzger
J. Differential Geom. 91(1): 81-102 (May 2012). DOI: 10.4310/jdg/1343133701

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.

Citation

Download Citation

Michael Eichmair. Jan Metzger. "On Large Volume Preserving Stable CMC Surfaces in Initial Data Sets." J. Differential Geom. 91 (1) 81 - 102, May 2012. https://doi.org/10.4310/jdg/1343133701

Information

Published: May 2012
First available in Project Euclid: 24 July 2012

zbMATH: 1278.53062
MathSciNet: MR2944962
Digital Object Identifier: 10.4310/jdg/1343133701

Rights: Copyright © 2012 Lehigh University

JOURNAL ARTICLE
22 PAGES


SHARE
Vol.91 • No. 1 • May 2012
Back to Top