Abstract
Abstract We extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.
Highlights
We complement in this paper our recent work [5] on the characterization of the leaves of the canonical foliation as the unique large closed embedded stable constant mean curvature surfaces in strongly asymptotically flat Riemannian 3-manifolds
We extend here the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres developed by S
Metzger and the second-named author [6], we extended this characterization further under the additional assumption that the scalar curvature of .M; g/ is non-negative in the following way: Choose a point p 2 M
Summary
We complement in this paper our recent work [5] on the characterization of the leaves of the canonical foliation as the unique large closed embedded stable constant mean curvature surfaces in strongly asymptotically flat Riemannian 3-manifolds. 1 to fully understand large stable constant mean curvature spheres in .M; g/ To study this final scenario and to investigate whether the assumption of additional homogeneity in the expansion of the metric (1.3) off of Schwarzschild in Theorem 1.2 is really necessary, we revisit in this paper the Lyapunov–Schmidt analysis carried out in [3]. With non-negative scalar curvature that is smoothly asymptotic to Schwarzschild of mass m > 0 in the sense that for all multi-indices I , and which contains a sequence 1†ko kD1 of outlying stable constant mean curvature spheres with. 1 along with all derivatives, where Tij is homogeneous of degree 2, and which contains a sequence 1†ko kD1 of outlying stable constant mean curvature spheres with. We note that there is a large body of work on stable constant mean curvature spheres in general asymptotically flat Riemannian 3-manifolds. We refer the reader to [5, Section 2.1] for an overview and references to results in this direction
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