On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

AbstractWe 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.

Download Full-text

4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500949 ◽

2020 ◽

pp. 1950094

Author(s):

Yaohua Wang

Keyword(s):

Scalar Curvature ◽

Upper Bound ◽

Mean Curvature ◽

Constant Mean Curvature ◽

Outer Boundary ◽

Compact Manifolds ◽

Manifolds With Boundary ◽

Flat Extension ◽

Asymptotically Flat ◽

3 Dimensional

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

Download Full-text

Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500656 ◽

2017 ◽

Vol 20 (01) ◽

pp. 1650065

Author(s):

Shiguang Ma

Keyword(s):

Riemannian Manifolds ◽

Stability Condition ◽

Mean Curvature ◽

Constant Mean Curvature ◽

Asymptotically Flat ◽

Cmc Surfaces ◽

Ode Method ◽

Nonlinear Ode ◽

The Stability

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 [Formula: see text]. 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.

Download Full-text